Remarks on L^{p}-Boundedness of Wave Operators for

Schrödinger Operators with Threshold Singularities

K. Yajima ^{1}

Received: February 24, 2016

Communicated by Heinz Siedentop

Abstract. We consider the continuity property in Lebesgue spaces L^{p}(ℝ^{m}) of the wave operators W_{±} of scattering theory for Schrödinger operators H = −Δ + V on ℝ^{m}, |V (x)| ≤ C⟨x⟩^{−δ} for some δ > 2 when H is of exceptional type, i.e. N = {u ∈ ⟨x⟩^{s}L^{2}(ℝ^{m}): (1 + (−Δ)^{−1}V)u = 0} ≠ {0} for some 1/2 < s < δ − 1/2. It has recently been proved by Goldberg and Green for m ≥ 5 that W_{±} are in general bounded in L^{p}(ℝ^{m}) for 1 ≤ p < m/2, for 1 ≤ p < m if all ϕ ∈ N satisfy ∫_{ℝ^{m}} V ϕdx = 0 and, for 1 ≤ p < ∞ if ∫_{ℝ^{m}} x_{i}V ϕdx = 0, i = 1, …, m in addition. We make the results for p > m/2 more precise and prove in particular that these conditions are also necessary for the stated properties of W_{±}. We also prove that, for m = 3, W_{±} are bounded in L^{p}(ℝ^{3}) for 1 < p < 3 and that the same holds for 1 < p < ∞ if and only if all ϕ ∈ N satisfy ∫_{ℝ^{3}} V ϕdx = 0 and ∫_{ℝ^{3}} x_{i}V ϕdx = 0, i = 1, 2, 3, simultaneously.

2010 Mathematics Subject Classification: 35P25, 81U05, 47A40.

1 Introduction

Let H_{0} = −Δ be the free Schrödinger operator on the Hilbert space H = L^{2}(ℝ^{m}) with domain D(H_{0}) = {u ∈ H: − Δu ∈ H} and H = H_{0} + V, V being the multiplication operator with the real measurable function V (x) which satisfies

|V (x)| ≤ C⟨x⟩^{−δ} for some δ > 2, ⟨x⟩ = (1 + |x|^{2})^{\frac{1}{2}} .  (1.1)

Then, H is selfadjoint in H with a core C_{0}^{∞}(ℝ^{m}) and it satisfies the following properties (see e.g. [18, 19, 21, 22, 23]):

1Supported by JSPS grant in aid for scientific research No. 22340029

Documenta Mathematica 21 (2016) 391–443

(i) The spectrum σ(H) of H consists of the absolutely continuous (AC for short) part [0, ∞) and a finite number of non-positive eigenvalues of finite multiplicities.

We write H_{ac}(H) for the AC spectral subspace of H for H, H_{ac} for the part of H in H_{ac}(H) and P_{ac}(H) for the orthogonal projection onto H_{ac}(H).

(ii) Wave operators W_{±} = lim_{t→±∞} e^{itH}e^{−itH_{0}} defined by strong limits exist and are complete, viz. Image W_{±} = H_{ac}(H). They are unitary from H onto H_{ac}(H) and intertwine H_{ac} and H_{0}. Hence, for Borel functions f,

f(H)P_{ac}(H) = W_{±}f(H_{0})W_{±}^{∗}.  (1.2)

If follows that various mapping properties of f(H)P_{ac} may be deduced from those of f(H_{0}) if the corresponding ones of W_{±} are known. In particular, if W_{±} ∈ B(L^{p}(ℝ^{m})) for 1 ≤ p_{1} ≤ p ≤ p_{2} < ∞, then W_{±}^{∗} ∈ B(L^{q}(ℝ^{m})) for q_{2} ≤ q ≤ q_{1}, 1/p_{j} + 1/q_{j} = 1, j = 1, 2, and

∥f(H)P_{ac}(H)∥_{B(L^{q},L^{p})} ≤ C_{pq}∥f(H_{0})∥_{B(L^{q},L^{p})},  (1.3)

for these p and q with C_{pq} which are independent of f. We define the Fourier and the conjugate Fourier transforms Fu(ξ) and F^{∗}u(ξ) respectively by

Fu(ξ) = ∫_{ℝ^{m}} e^{−ixξ}u(x)dx and F^{∗}u(ξ) = \frac{1}{(2π)^{m}} ∫_{ℝ^{m}} e^{ixξ}u(x)dx.

We also write û(ξ) for Fu(ξ). The intertwining property (1.2) may be made more precise. Wave operators W_{±} are transplantations ([24]) of the complete set of (generalized) eigenfunctions {e^{ixξ}: ξ ∈ ℝ^{m}} of −Δ by those of out-going and in-coming scattering eigenfunctions {φ_{±}(x, ξ): ξ ∈ ℝ^{m}} of H = −Δ + V ([19]):

W_{±}u(x) = F_{±}^{∗}Fu(x) = \frac{1}{(2π)^{d}} ∫_{ℝ^{d}} φ_{±}(x, ξ)û(ξ)dξ,

where F_{±} and F_{±}^{∗} are the generalized Fourier transforms associated with {φ_{±}(x, ξ): ξ ∈ ℝ^{m}} and the conjugate ones defined respectively by

F_{±}u(ξ) = ∫_{ℝ^{d}} \overline{φ_{±}(x, ξ)}u(x)dx, F_{±}^{∗}u(ξ) = \frac{1}{(2π)^{m}} ∫_{ℝ^{d}} φ_{±}(x, ξ)u(x)dx.

They satisfy F_{±}^{∗}F_{±}u = u for u ∈ H_{ac}(H) and, F_{±}F_{±}^{∗}u = u for u ∈ L^{2}(ℝ^{m}). We define F (D) ≡ F^{∗}M_{F} F and F (D_{±}) ≡ F_{±}^{∗}M_{F} F_{±} for Borel functions F on ℝ^{m} where M_{F} is the multiplication with F (ξ). Then,

F (D_{±}) = W_{±}F (D)W_{±}^{∗}u, u ∈ H_{ac}(H)

and W_{±} transplant estimates for F (D) in L^{p}-spaces to F (D_{±}).

Remarks on L^{p}-Boundedness of Wave Operators for … 393

In this paper we are interested in the problem whether or not W_{±} are bounded in L^{p}(ℝ^{m}). This will almost automatically imply the same property in Sobolev spaces W^{k,p}(ℝ^{m}) = {u ∈ L^{p}(ℝ^{m}): ∂^{α}u ∈ L^{p}(ℝ^{m})} for integers 0 ≤ k ≤ 2 (see Section 7 of [8]). There is now a large literature on this problem ([3, 4, 6, 8, 27, 31, 15, 16, 29, 33]) and it is well known that the answer depends on the spectral properties of H at 0, the bottom of the AC spectrum of H. We define

E = {u ∈ H^{2}(ℝ^{m}): (−Δ + V)u = 0},  (1.4)

the eigenspace of H with eigenvalue 0 and, for 1/2 < s < δ − 1/2,

N = {u ∈ ⟨x⟩^{s}L^{2}(ℝ^{m}) : (1 + (−Δ)^{−1}V)u = 0} = 0.  (1.5)

Functions ϕ in N satisfy −Δϕ + V ϕ = 0 for x ∈ ℝ^{m}. The space N is finite dimensional, independent of 1/2 < s < δ − 1/2, E ⊂ N and, if m ≥ 5, E = N ([14]). The operator H is said be of generic type if N = {0} and of exceptional type otherwise. When H is of generic type, we have rather satisfactory results (though there is much space for improving conditions on V) and it has been proved that W_{±} are bounded in L^{p}(ℝ^{m}) for all 1 ≤ p ≤ ∞ if m ≥ 3 and, for all 1 < p < ∞ if m = 1 and m = 2 under various smoothness and decay at infinity assumptions on V (see [4] for the best result when m = 3); but they are in general not bounded in L^{1}(ℝ^{1}) or L^{∞}(ℝ^{1}) when m = 1 ([27]). When H is of exceptional type, it is long known that the same results hold when m = 1 (see [27, 3, 6]). For higher dimensions m ≥ 3, it is first shown ([33, 8]) that W_{±} are bounded in L^{p}(ℝ^{m}) for 3/2 < p < 3 if m = 3 and for \frac{m}{m−2} < p < \frac{m}{2} if m ≥ 5, which is subsequently extended to 1 < p < 3 for m = 3 and 1 < p < m/2 for m ≥ 5 ([34]). Then, recently, Goldberg and Green ([10]) have substantially improved these results by proving the following theorem for m ≥ 5. In what follows in this paper, we assume m ≥ 3 and V satisfies the following assumption. The constant m_{∗} is defined by

m_{∗} = (m − 1)/(m − 2).

Assumption 1.1. V is a real valued measurable function such that

(1) F(⟨x⟩^{2σ}V) ∈ L^{m_{∗}} for some σ > 1/m_{∗}.

(2) |V (x)| ≤ C⟨x⟩^{−δ} for some δ > \{ \begin{matrix}m + 4, & if 3 ≤ m ≤ 7, \\ m + 3, & if m ≥ 8 \end{matrix} and C > 0.

The condition (1) requires certain smoothness on V. We write ⟨u, v⟩ = ∫_{ℝ^{m}} \overline{u(x)}v(x)dx and define subspaces E_{1} ⊂ E_{0} ⊂ N respectively by

E_{0} = {ϕ ∈ N : ⟨V, ϕ⟩ = 0}, E_{1} = {ϕ ∈ E_{0} : ⟨xV, ϕ⟩ = 0},  (1.6)

where ⟨xV, ϕ⟩ = 0 means ⟨x_{i}V, ϕ⟩ = 0 for all 1 ≤ i ≤ m. We have dim N /E_{0} ≤ 1, E_{0} = E if m = 3 and N = E if m ≥ 5.

Theorem 1.2 (Goldberg-Green). Suppose that V satisfies Assumption 1.1 and that H is of exceptional type. Then, if m ≥ 5, W_{±} are bounded in L^{p}(ℝ^{m}) for 1 ≤ p < m/2. They are bounded in L^{p}(ℝ^{m}) also for 1 ≤ p < m if N = E_{0} and for 1 ≤ p < ∞ if N = E_{1}.

In this paper, we show following theorems which in particular prove the corresponding result for m = 3 and that the conditions N = E_{0} and N = E_{1} of Theorem 1.2 are also necessary for the stated properties of W_{±} respectively. We write P, P_{0} and P_{1} for the orthogonal projections onto E, E_{0} and E_{1} respectively. Because (−Δ)^{−1}V is a real operator, we may take the bases of N, E_{0} and E_{1} which consist of real functions and P, P_{0} and P_{1} are real operators: For the conjugation (Cu)(x) = \overline{u(x)},

C^{−1}P C = P, C^{−1}P_{0}C = P, C^{−1}P_{1}C = P_{1}.  (1.7)

We state results for m = 3, m = 5 and m ≥ 6 separately.

Theorem 1.3. Let m = 3. Suppose that V satisfies Assumption 1.1 and that H is of exceptional type. Then:

(1) W_{±} are bounded in L^{p}(ℝ^{3}) for 1 < p < 3.

(2) For 3 < p < ∞, there exists a constant C such that

∥(W_{±} ± aφ ⊗ |D|^{−1}V φ + P)u∥_{L^{p}} ≤ C∥u∥_{L^{p}},  (1.8)

where φ is the real function defined by (3.13) (the canonical resonance), a = 4πi|⟨V, φ⟩|^{−2} and P may be replaced by P ⊖ P_{1}.

(3) If W_{±} are bounded in L^{p}(ℝ^{3}) for some 3 < p < ∞, then N = E_{1}. In this case they are bounded in L^{p}(ℝ^{3}) for all 1 < p < ∞.

Theorem 1.4. Let m = 5. Suppose that V satisfies Assumption 1.1 and that H is of exceptional type. Then:

(1) W_{±} are bounded in L^{p}(ℝ^{5}) for 1 < p < 5/2.

(2) For 5/2 < p < 5, there exists a constant C such that

∥(W_{±} ± a_{0}(|D|^{−1}V φ) ⊗ φ + \frac{P}{2} ) u∥_{L^{p}} ≤ C∥u∥_{L^{p}},  (1.9)

where φ = P V, V being considered as a function, a_{0} = i/(24π^{2}) and P may be replaced by P ⊖ P_{0}. If W_{±} are bounded in L^{p}(ℝ^{5}) for some \frac{5}{2} < p < 5, then N = E_{0}. In this case they are bounded in L^{p}(ℝ^{5}) for all 1 < p < 5.

(3) By virtue of (1) and (2), the condition E = E_{0} is necessary for W_{±} to be bounded in L^{p}(ℝ^{5}) for some p > 5. Suppose E = E_{0}. Then,

∥(W_{±} + P) u∥_{L^{p}} ≤ C∥u∥_{L^{p}}  (1.10)

for a constant C, where P = P_{0} may be replaced by P_{0} ⊖ P_{1}. If W_{±} are bounded in L^{p}(ℝ^{5}) for some p > m, then N = E_{1}. In this case they are bounded in L^{p}(ℝ^{5}) for all 1 < p < ∞.

Theorem 1.5. Let m ≥ 6. Suppose that V satisfies Assumption 1.1 and that H is of exceptional type. Then:

(1) W_{±} are bounded in L^{p}(ℝ^{m}) for 1 < p < m/2.

(2) For \frac{m}{2} < p < m, there exists a constant C > 0 such that

∥(W_{±} + D_{m}P)u∥_{L^{p}} ≤ C_{p}∥u∥_{L^{p}},  (1.11)

where P may be replaced by P ⊖ P_{0} and

D_{m} = {

\frac{Γ (\frac{m−2}{2})}{\sqrt{π}Γ (\frac{m−1}{2})},  m is odd, (1.12)

\frac{2^{m}Γ (\frac{m}{2})}{\sqrt{π}Γ (\frac{m−1}{2})} ∫_{1}^{∞}(x^{2} + 1)^{−(m−1)}dx,  m is even. (1.13)

If W_{±} are bounded in L^{p}(ℝ^{m}) for some m/2 < p < m then, E = E_{0}. In this case they are bounded in L^{p}(ℝ^{m}) for all 1 < p < m

(3) Suppose E = E_{0}. Let m < p < ∞. Then, for a constant C_{p},

∥(W_{±} + P)u∥ ≤ C_{p}∥u∥_{L^{p}},  (1.14)

where P may be replaced by P_{0} ⊖ P_{1}. If W_{±} are bounded in L^{p}(ℝ^{m}) for some p > m, then E = E_{1}. In this case they are bounded in L^{p}(ℝ^{m}) for all 1 < p < ∞.

Remark 1.6. (1) The integral in (1.13) may be computed explicitly:

∫_{1}^{∞}(x^{2} + 1)^{−(m−1)}dx = \frac{Γ (m − \frac{3}{2})}{4Γ(m − 1)} (\sqrt{π} − ∑_{j=1}^{m−2} \frac{Γ(j)2^{−j+1}}{Γ (j + \frac{1}{2})} ) .  (1.15)

(2) There are examples of V such that E_{1} = E_{0} ⊊ N, E_{1} ⊊ E_{0} = N and E_{1} ⊊ E_{0} ⊊ N (see Example 8.4 of [13]).

(3) Murata’s result (Theorem 1.2 of [20]) also implies that, if N ≠ 0, W_{±} are not in general bounded in L^{p}(ℝ^{m}) for p > 3 if m = 3 and for p > \frac{m}{2} if m ≥ 5.

The rest of the paper is devoted to the proof of Theorems. In spite that substantial part of Theorems 1.4 and 1.5 overlaps with Theorem 1.2 and that they miss critically important L^{1}-boundedness, we present the proof of Theorems which is very different from the one by Goldberg and Green ([10]). Our proof heavily uses harmonic analysis machinery, which produces sharper results for larger p’s, however, at the same time, prevents us from reaching end points p = 1 and p = ∞. We prove the theorems only for W_{−} since conjugation changes the direction of time, viz. C^{−1}e^{−itH}C = e^{itH}, and

W_{+} = C^{−1}W_{−}C.  (1.16)

We use the following notation and conventions: The ℓ-th derivative of f(x), x ∈ ℝ is denoted by f^{(ℓ)}(x). Σ = 𝕊^{m−1} = {x: x_{1}^{2} + ⋯ + x_{m}^{2} = 1} is the unit sphere in ℝ^{m} and ω_{m−1} = 2π^{\frac{m}{2}} /Γ (\frac{m}{2}) is its area. The coupling and the inner product are anti-linear with respect to the first component,

(u, v) = ⟨u, v⟩ = ∫_{ℝ^{n}} \overline{u(x)}v(x)dx,

in accordance with the interchangeable notaion for the rank 1 operator

|u⟩⟨v| = u ⊗ v : ϕ ↦ u⟨v, ϕ⟩.

This notation is used also when v is in a certain function space and u in its dual space.

f ≤_{| · |} g means |f| ≤ |g|.

For Banach spaces X and Y, B(X, Y) is the Banach space of bounded operators from X to Y and B(X) = B(X, X); B_{∞}(X, Y) and B_{∞}(X) are spaces of compact operators; and the dual space B(X, ℂ) of X is denoted by X^{∗}. The identity operators in various Banach spaces are indistinguishably denoted by 1. For 1 ≤ p ≤ ∞, ∥u∥_{p} = ∥u∥_{L^{p}} is the norm of L^{p}(ℝ^{m}) and p′ is its dual exponent, 1/p + 1/p′ = 1. When p = 2, we often omit p and write ∥u∥ for ∥u∥_{2}. We interchangeably write L_{w}^{p}(ℝ^{m}) or L^{p,∞}(ℝ^{m}) for weak-L^{p} spaces and ∥u∥_{p,w} or ∥u∥_{p,∞} for their norms. For s ∈ ℝ,

L_{s}^{2} = ⟨x⟩^{−s}L^{2} = L^{2}(ℝ^{m}, ⟨x⟩^{2s}dx), H^{s}(ℝ^{m}) = FL_{s}^{2}(ℝ^{m})

are the weighted L^{2} spaces and Sobolev spaces. The space of rapidly decreasing functions is denoted by S(ℝ^{m}). We denote the resolvents of H and H_{0} respectively by

R(z) = (H − z)^{−1}, R_{0}(z) = (H_{0} − z)^{−1}.

We parameterize z ∈ ℂ\[0, ∞) as z = λ^{2} by λ ∈ ℂ^{+}, the open upper half plane of ℂ, so that the positive and the negative parts of the boundary {λ: ± λ ∈ (0, ∞)} are mapped onto the upper and the lower edges of the positive half line {z ∈ ℂ: z > 0}. We define

G(λ) = R(λ^{2}), G_{0}(λ) = R_{0}(λ^{2}), λ ∈ ℂ^{+}.

These are B(H)-valued meromorphic functions of λ ∈ ℂ^{+} and the limiting absorption principle [19] (LAP for short) asserts that, when considered as B(⟨x⟩^{−s}L^{2}, ⟨x⟩^{t}L^{2})-valued functions for s, t > \frac{1}{2} and s + t > 2, G_{0}(λ) has Hölder continuous extensions to its closure \overline{ℂ}^{+} = \{z : ℑz ≥ 0\}. The same is true also for G(λ), but, if H is of exceptional type, it has singularities at λ = 0. In what follows z^{\frac{1}{2}} is the branch of square root of z cut along the negative real axis such that z^{\frac{1}{2}} > 0 when z > 0. The plan of the paper is as follows: In section 2, we record some results most of which are well known and which we use in the sequel. They include:

• Formulas for the integral kernel of G_{0}(λ) as exponential-polynomials in odd dimensions or their superpositions in even dimensions.

• Representation of ⟨ψ|(G_{0}(λ) − G_{0}(−λ))u⟩ as the linear combination of Fourier transforms of r^{j+1}M(r, \overline{ψ} ∗ ǔ), M(r, f) being the average of f over the sphere of radius r centered at the origin.

• The Muckenhaupt weighted inequality and examples of A_{p}-weights.

In section 3, we recall and improve results of [33] and [8] on the behavior as λ → 0 of (1 + G_{0}(λ)V)^{−1} and reduce the problem to the L^{p}-boundedness of

Z_{s}u = −\frac{1}{πi} ∫_{0}^{∞} G_{0}(λ)V S(λ)(G_{0}(λ) − G_{0}(−λ))uλF (λ)dλ  (1.17)

where S(λ) is the singular part of the expansion of (1 + G_{0}(λ)V)^{−1} at λ = 0 and F ∈ C_{0}^{∞}(ℝ) is such that F (λ) = 1 near λ = 0. We prove Theorem 1.3 in Section 4, Theorem 1.4 for odd dimensions m ≥ 5 in Section 5 and for even dimensions in Section 6. We explain the basic strategy of the proof at the end of §4.1 after most of basic ideas appears in the simplest form. Acknowledgment The part of this work was carried out while the author was visiting Aalborg and Aarhus universities in summers of 2014 and 2015 respectively. He would like to express his sincere gratitude to both institutions for the hospitality, to Arne Jensen, Jacob Schach Møller and Erik Skibsted in particular. He thanks Professors Goldberg and Green for sending us their manuscript [10] before posting on the arXiv-math, Shin Nakano for computing the recursion formula for (1.15) and Fumihiko Nakano for bringing [12] to his attention.

2 Preliminaries

In this section we record some well known results which we use in what follows.

2.1 Integral kernel of the free resolvent

For m ≥ 2, resolvent G_{0}(λ) for ℑλ ≥ 0 is the convolution with

G_{0}(λ, x) = \frac{e^{iλ|x|}}{2(2π)^{\frac{m−1}{2}} Γ (\frac{m−1}{2}) |x|^{m−2}} ∫_{0}^{∞} e^{−t}t^{\frac{m−3}{2}} (\frac{t}{2} − iλ|x|)^{\frac{m−3}{2}} dt (2.1)

Documenta Mathematica 21 (2016) 391–443

([28]). When m ≥ 3 is odd, it is an exponential polynomial like function.

Lemma 2.1. Let m ≥ 3 be odd. Then:

G_{0}(λ, x) = ∑_{j=0}^{(m−3)/2} C_{j} \frac{(λ|x|)^{j}e^{iλ|x|}}{|x|^{m−2}} with C_{j} = \frac{(−i)^{j}(m − 3 − j)!}{2^{m−1−j}π^{\frac{m−1}{2}} j!(\frac{m−3}{2} − j)!}.

(2.2)

The constant C_{0} may also be written as C_{0} = (m − 2)^{−1}ω_{m−1}^{−1} and

iC_{0} + C_{1} = 0, when m ≥ 5.  (2.3)

If m is even, the structure of G_{0}(λ, x) is more complex and this makes the analysis harder. For partly circumventing the difficulty we express G_{0}(λ, x) as a superposition of exponential-polynomial like functions of the form (2.2). This will allow a part of the proof for even dimensions to go in parallel with the odd dimensional cases. We set

ν = \frac{m − 2}{2}.

Define operators T_{j}^{(a)}, j = 0, …, ν for superposing over parameter a > 0 by

T_{j}^{(a)}[f(x, a)] = C_{m,j}ω_{m−1} ∫_{0}^{∞}(1 + a)^{−(2ν−j+ \frac{1}{2} )}f(x, a)\frac{da}{\sqrt{a}},  (2.4)
C_{m,j}ω_{m−1} = (−2i)^{j} \frac{Γ (2ν − j + \frac{1}{2})}{(m − 2)!\sqrt{π}} (\begin{matrix}ν \\ j \end{matrix}) .  (2.5)

The factor ω_{m−1} is added for shorting some formulas below (see (2.18)).

Lemma 2.2. If m ≥ 4 is even, then we have

G_{0}(λ, x) = ∑_{j=0}^{ν} ω_{m−1}^{−1}T_{j}^{(a)} [e^{iλ|x|(1+2a)} \frac{(λ|x|)^{j}}{|x|^{m−2}} ] .  (2.6)

Proof. Let C_{m∗} = 2^{m−1}π^{\frac{m−1}{2}} Γ (\frac{m−1}{2}). In the formula (2.1):

G_{0}(λ, x) = \frac{e^{iλ|x|}}{C_{m∗}|x|^{m−2}} ∫_{0}^{∞} e^{−t}t^{\frac{m−3}{2}} (t − 2iλ|x|)^{\frac{m−3}{2}} dt,  (2.7)

write (t − 2iλ|x|)^{\frac{m−3}{2}} = (t − 2iλ|x|)^{ν}(t − 2iλ|x|)^{− \frac{1}{2}} , expand (t − 2iλ|x|)^{ν} via the binomial formula and use the identity

z^{− \frac{1}{2}} = \frac{1}{\sqrt{π}} ∫_{0}^{∞} e^{−az} a^{− \frac{1}{2}} da, ℜ z > 0  (2.8)

for (t − 2iλ|x|)^{− \frac{1}{2}} . The right hand side of (2.7) becomes

∑_{j=0}^{ν} \frac{(−2i)^{j}}{\sqrt{π}C_{m∗}} (\begin{matrix}ν \\ j \end{matrix}) ∫∫_{ℝ_{+}^{2}} e^{−(1+a)t}t^{2ν−j} (e^{iλ|x|(1+2a)} \frac{(λ|x|)^{j}}{|x|^{m−2}} ) \frac{dt}{\sqrt{t}} \frac{da}{\sqrt{a}}.

The integral converges absolutely if m ≥ 4 and we obtain (2.6) after performing the integral with respect to t.

2.2 Spectral measure of H_{0}

. The spectral measure of H_{0} = −Δ is AC and Stone’s theorem implies that the spectral projection E_{0}(dμ) is given for μ = λ^{2}, λ > 0 by

E_{0}(dμ) = \frac{1}{2πi}(R_{0}(μ + i0) − R_{0}(μ − i0))dμ = \frac{1}{iπ} (G_{0}(λ) − G_{0}(−λ))λdλ.

Lemma 2.3. Let m ≥ 3 and u, v ∈ (L^{1} ∩ L^{2})(ℝ^{m}). Then, both sides of the following equation can be continuously extended to λ = 0 and

λ^{−1}⟨v, (G_{0}(λ) − G_{0}(−λ))u⟩ = ⟨|D|^{−1}v, (G_{0}(λ) − G_{0}(−λ))u⟩, λ ≥ 0. (2.9)

For bounded continuous functions f on ℝ we have for λ ≥ 0,

f(λ)⟨v, (G_{0}(λ)u − G_{0}(−λ))u⟩ = ⟨v, (G_{0}(λ)u − G_{0}(−λ))f(|D|)u⟩. (2.10)

Proof. For u, v ∈ (L^{1} ∩ L^{2})(ℝ^{m}) we have

⟨v, (G_{0}(λ) − G_{0}(−λ))u⟩ = \frac{λ^{m−2}i}{2(2π)^{m−1}} ∫_{Σ} \overline{v̂(λω)}û(λω)dω,  (2.11)

where Σ = 𝕊^{m−1}. It follows, since { \atop |D|^{−1}v}(λω) = λ^{−1}v̂(λω), λ > 0, that

⟨|D|^{−1}v, (G_{0}(λ) − G_{0}(−λ))u⟩ = \frac{λ^{m−3}i}{2(2π)^{m−1}} ∫_{Σ} \overline{v̂(λω)}û(λω)dω.  (2.12)

The right side extends to a continuous function of λ ≥ 0 when m ≥ 3 and (2.9) follows by comparing (2.11) and (2.12). Eqn. (2.10) likewise follows.

We define the spherical average of a function f on ℝ^{m} by

M(r, f) = \frac{1}{ω_{m−1}} ∫_{Σ} f(rω)dω, for all r ∈ ℝ.  (2.13)

We often write M_{f}(r) = M(r, f). We have M_{f}(−r) = M_{f}(r) and Hölder’s inequality implies

(\frac{1}{ω_{m−1}} ∫_{0}^{∞} |M_{f}(r)|^{p}r^{m−1}dr)^{1/p} ≤ ∥f∥_{p}, 1 ≤ p ≤ ∞.  (2.14)

For an even function M(r) of r ∈ ℝ, define M̃(ρ) by

M̃(ρ) = ∫_{ρ}^{∞} rM(r)dr (= − ∫_{−∞}^{ρ} rM(r)dr) .  (2.15)

Lemma 2.4. Suppose M(r) = M(−r) and ⟨r⟩^{2}M(r) is integrable. Then,

∫_{ℝ} e^{−irλ}rM(r)dr = \frac{λ}{i} ∫_{ℝ} e^{−irλ}M̃(r)dr, ∫_{ℝ} M̃(r)dr = ∫_{ℝ} r^{2}M(r)dr. (2.16)

Proof. Since rM(r) = −M̃(r)′, integration by parts gives the first equation. We differentiate both sides of the first and set λ = 0. The second follows.

We denote ǔ(x) = u(−x), x ∈ ℝ^{m}. (The sign ǔ will be reserved for this purpose and will not be used to denote the conjugate Fourier transform.)

Representation formula for odd dimensions.

Lemma 2.5. Let m ≥ 3 be odd and u, ψ ∈ C_{0}^{∞}(ℝ^{m}). Define c_{j} = ω_{m−1}C_{j}, 1 ≤ j ≤ \frac{m−3}{2}, where C_{j} are the constants in (2.2). Then, for λ > 0 we have

⟨ψ, (G_{0}(λ) − G_{0}(−λ))u⟩ = ∑_{j=0}^{\frac{m−3}{2}} c_{j}(−1)^{j+1}λ^{j} ∫_{ℝ} e^{−iλr}r^{1+j}M_{\overline{ψ}∗ǔ}(r)dr. (2.17)

Proof. We compute ⟨ψ, G_{0}(λ)u⟩ by using the integral kernel (2.2) of G_{0}(λ). Change the order of integration and use polar coordinates. Then,

⟨ψ, G_{0}(λ)u⟩ = ∑_{j=0}^{\frac{m−3}{2}} C_{j} ∫_{ℝ^{m}} \overline{ψ(x)} (∫_{ℝ^{m}} \frac{λ^{j}e^{iλ|y|}u(x − y)}{|y|^{m−2−j}}dy) dx

= ∑_{j=0}^{\frac{m−3}{2}} C_{j} ∫_{ℝ^{m}} \frac{λ^{j}e^{iλ|y|}(\overline{ψ} ∗ ǔ)(y)}{|y|^{m−2−j}}dy = ∑_{j=0}^{\frac{m−3}{2}} c_{j} ∫_{0}^{∞} λ^{j}e^{iλr}r^{1+j}M_{\overline{ψ}∗ǔ}(r)dr.

Since M_{\overline{ψ}∗ǔ}(r) is even, change of variable r by −r yields

−⟨ψ, G_{0}(−λ)u⟩ = ∑_{j=0}^{\frac{m−3}{2}} c_{j} ∫_{−∞}^{0} λ^{j}e^{iλr}r^{1+j}M_{\overline{ψ}∗ǔ}(r)dr.

Add both sides of last two equations and change r by −r.

Representation formula for even dimensions. If m is even, we have the analogue of (2.17). For a function M(r) on ℝ and a > 0, define

M^{a}(r) = M((1 + 2a)^{−1}r).

Lemma 2.6. Let m ≥ 2. Let u, ψ ∈ C_{0}^{∞}(ℝ^{m}). Then

⟨ψ, (G_{0}(λ) − G_{0}(−λ))u⟩ = ∑_{j=0}^{ν}(−1)^{j+1}T_{j}^{(a)} [\frac{λ^{j}F(r^{j+1}M_{\overline{ψ}∗ǔ}^{a})(λ)}{(1 + 2a)^{j+2}}] , (2.18)

For j = 0, −T_{0}^{(a)} [\frac{F(rM_{\overline{ψ}∗ǔ}^{a})(λ)}{(1 + 2a)^{2}}] = iT_{0}^{(a)} [\frac{λ(F{ \atop M_{\overline{ψ}∗ǔ}^{a}})(λ)}{(1 + 2a)^{2}} ] .  (2.19)

Proof. Define B_{j}(λ, r, a) = e^{iλr(1+2a)}(λr)^{j}r^{−(m−2)} and

B_{j}(λ, a)u(x) = ∫_{ℝ^{m}} B_{j}(λ, |y|, a)u(x − y)dy, j = 0, …, ν.

Then, (2.6) and change of the order of integrations imply

⟨ψ, (G_{0}(λ) − G_{0}(−λ))u⟩ = ∑_{j=0}^{ν} \frac{T_{j}^{(a)}}{ω_{m−1}} [⟨ψ, (B_{j}(λ, a) − B_{j}(−λ, a))u⟩] . (2.20)

We have, as in odd dimensions, that for u ∈ S(ℝ^{m}) and ψ ∈ L^{1}(ℝ^{m})

⟨ψ, B_{j}(λ, a)u⟩ = ∫_{ℝ^{m}} (∫_{ℝ^{m}} \overline{ψ(x)}B_{j}(λ, |y|, a)u(x − y)dy) dx

= ∫_{ℝ^{m}} B_{j}(λ, |y|, a)(\overline{ψ} ∗ ǔ)(y)dy = ω_{m−1} ∫_{0}^{∞} e^{i(1+2a)λr}(λr)^{j}rM_{\overline{ψ}∗ǔ}(r)dr.

Replacing λ to −λ and changing the variable r to −r, we have

−⟨ψ, B_{j}(−λ, a)u⟩ = ω_{m−1} ∫_{−∞}^{0} e^{i(1+2a)λr}(λr)^{j}rM_{\overline{ψ}∗ǔ}(r)dr,

where we used that M_{\overline{ψ}∗ǔ}(−r) = M_{\overline{ψ}∗ǔ}(r). Adding these two yields

⟨ψ, (B_{j}(λ, a) − B_{j}(−λ, a))u⟩ = ω_{m−1} ∫_{ℝ} e^{i(1+2a)λr}(λr)^{j}rM_{\overline{ψ}∗ǔ}(r)dr. (2.21)

Change r to −r in the right of (2.21), plug the result with (2.20) and, at the end, change the variable r to −r/(1 + 2a). Then, (2.21) becomes

\frac{(−1)^{j+1}ω_{m−1}}{(1 + 2a)^{j+2}} ∫_{ℝ} e^{−iλr}λ^{j}r^{j+1}M_{\overline{ψ}∗ǔ}^{a}(r)dr = \frac{(−1)^{j+1}ω_{m−1}λ^{j}}{(1 + 2a)^{j+2}} F(r^{j+1}M_{\overline{ψ}∗ǔ}^{a})(λ)

and (2.18) follows. If we use the first of (2.16), the right of the last equation for j = 0 becomes

\frac{iλ(F{ \atop M_{\overline{ψ}∗ǔ}^{a}})(λ)}{(1 + 2a)^{2}}ω_{m−1}

and we obtain (2.19).

2.3 Some results from harmonic analysis.

The following lemma on weighted inequality (cf. [11], Chapter 9) plays crucial role in this paper. We let 1 < p < ∞ in this subsection.

Lemma 2.7. The weight function |r|^{a} is an A_{p} weight on ℝ if and only if −1 < a < p − 1. The Hilbert transform H̃ and the Hardy-Littlewood maximal operator M are bounded in L^{p}(ℝ, w(r)dr) for A_{p} weights w(r).

Modifying the Hilbert transform H̃, we define

Hu(ρ) = \frac{(1 + H̃)u(ρ)}{2} = \frac{1}{2π} ∫_{0}^{∞} e^{irρ}û(r)dr.  (2.22)

We shall repeatedly use following A_{p} weights on ℝ^{1} to the operator MH:

|r|^{m−1−p(m−1)}, |r|^{m−1−2p}, |r|^{m−1−p} and |r|^{m−1},  (2.23)

respectively for 1 < p < \frac{m}{m−1}, \frac{m}{3} < p < \frac{m}{2}, \frac{m}{2} < p < m and m < p. For a function F (x) on ℝ^{m}, we say G(|x|) ∈ L^{1}(ℝ^{m}) is a radially decreasing integrable majorant (RDIM for short) of F if G(r) > 0 is decreasing and |F (x)| ≤ G(|x|) for a.e. x ∈ ℝ^{m}. The following lemma is well known (see e.g. [26], p.57).

Lemma 2.8. (1) A rapidly decreasing function F ∈ S(ℝ^{m}) has a RDIM.

(2) If F has a RDIM. then there is a constant C > 0 such that

|(F ∗ u)(t)| ≤ C(Mu)(t), t ∈ ℝ.  (2.24)

Lemma 2.9. For u and F ∈ L^{1}(ℝ) such that û, F̂ ∈ L^{1}(ℝ) we have

\frac{1}{2π} ∫_{0}^{∞} e^{iλρ}F (λ)û(λ)dλ = (F^{∗}F ∗ Hu)(ρ).  (2.25)

Proof. Let Θ(λ) = \{ \begin{matrix}1, for λ > 0 \\ 0, for λ ≤ 0 \end{matrix} . Then, the left side of (2.25) equals

\frac{1}{2π} ∫_{ℝ} e^{iλρ}F (λ)Θ(λ)û(λ)dλ = \frac{1}{2π} ∫_{ℝ} (∫_{ℝ} e^{iλ(ρ−ξ)}F^{∗}F (ξ)dξ) Θ(λ)û(λ)dλ

= ∫_{ℝ} F^{∗}F (ξ)F^{∗}{Θ(λ)û(λ)}(ρ − ξ)dξ = (F^{∗}F ∗ Hu)(ρ)

as desired.

3 Reduction to the low energy analysis

We write W_{−} = W in the sequel. When u ∈ ⟨x⟩^{−s}L^{2}, s > 1/2, W u may be expressed via the boundary values of resolvents (e.g. [19]):

W u = u − lim_{ε↓0,N↑∞} \frac{1}{πi} ∫_{ε}^{N} G(λ)V (G_{0}(λ) − G_{0}(−λ))uλdλ  (3.1)
= u − \frac{1}{πi} ∫_{0}^{∞} G(λ)V (G_{0}(λ) − G_{0}(−λ))uλdλ  (3.2)

Here the right of (3.1) is the Riemann integral of an ⟨x⟩^{t}L^{2}-valued continuous function where t > 1/2 is such that s + t > 2, the result belongs to L^{2}(ℝ^{m}) and the limit exists in L^{2}(ℝ^{m}), which we symbolically write as (3.2). We decompose W into the high and the low energy parts

W = W_{>} + W_{<} ≡ W Ψ(H_{0}) + W Φ(H_{0}),  (3.3)

by using cut off functions Φ ∈ C_{0}^{∞}(ℝ) and Ψ ∈ C^{∞}(ℝ) such that

Φ(λ^{2}) + Ψ(λ^{2}) ≡ 1, Φ(λ^{2}) = 1 near λ = 0 and Φ(λ^{2}) = 0 for |λ| > λ_{0} for a small constant λ_{0} > 0. We have proven in previous papers [33, 8] that, under Assumption 1.1, W_{>} is bounded in L^{p}(ℝ^{m}) for all 1 ≤ p ≤ ∞ if m ≥ 3 and we only need to study W_{<} = Φ(H_{0}) + Z where

Z = −\frac{1}{πi} ∫_{0}^{∞} G(λ)V (G_{0}(λ) − G_{0}(−λ))λΦ(H_{0})dλ.  (3.4)

Evidently Φ(H_{0}) ∈ B(L^{p}(ℝ^{m})) for all 1 ≤ p ≤ ∞ and we have only to study the operator Z defined by (3.4). Since δ > 2, the LAP (cf. Lemma 2.2 of [33]) implies that G_{0}(λ)V is a Hölder continuous function of λ ∈ ℝ with values in B_{∞}(L^{−s}) for any \frac{1}{2} < s < δ − \frac{1}{2} and, the absence of positive eigenvalues ([17]) implies that 1 + G_{0}(λ)V is invertible for λ > 0 (cf. [1]). It follows from the resolvent equation G(λ) = G_{0}(λ) − G_{0}(λ)V G(λ) that G(λ)V may be expressed in terms of G_{0}(λ)V:

G(λ)V = G_{0}(λ)V (1 + G_{0}(λ)V)^{−1} for λ ≠ 0  (3.5)

and it is locally Hölder continuous for λ ∈ ℝ \ {0} with values in B_{∞}(L_{−s}^{2}). Thus, we have the expression of Z in terms of the free resolvent G_{0}(λ):

Zu = −\frac{1}{πi} ∫_{0}^{∞} G_{0}(λ)V (1 + G_{0}(λ)V)^{−1}(G_{0}(λ) − G_{0}(−λ))λF (λ)udλ, (3.6)

where F (λ) = Φ(λ^{2}). If H is of generic type, Ker_{L_{−s}^{2}}(1 + G_{0}(0)V) = N = {0} for any \frac{1}{2} < s < δ − \frac{1}{2} and 1 + G_{0}(λ)V is invertible for λ in a neighborhood λ = 0 and both sides of (3.5) become Hölder continuous. We then have shown in [33, 8] that Z is bounded in L^{p}(ℝ^{m}) for all 1 ≤ p ≤ ∞ under Assumption 1.1.

3.1 Low energy behavior of (1 + G_{0}(λ)V)^{−1}.

If H is of exceptional type, (1 + G_{0}(λ)V)^{−1} becomes singular at λ = 0 and we describe its singularities here. Before doing so we recall some properties of functions in N. Recall ([25]) that for 0 < s < m :

|D|^{−s}u(x) = F^{∗}(|ξ|^{−s}û)(x) = \frac{Γ (\frac{m−s}{2})}{2^{s}π^{\frac{m}{2}} Γ (\frac{s}{2})} ∫_{ℝ^{m}} \frac{u(y)}{|x − y|^{m−s}} dy.  (3.7)

When s = 1 and s = 2, the constants in front of the integral respectively equal to π^{−1}ω_{m−2}^{−1} and C_{0} = (m − 2)^{−1}ω_{m−1}^{−1} of (2.2).

Lemma 3.1. (1) Functions ϕ in N satisfy ⟨x⟩^{−s}ϕ ∈ H^{2}(ℝ^{m}) ∩ C^{1}(ℝ^{m}) for any s > 1/2 and ∇ϕ is Hölder continuous. They satisfy the following asymptotic expansion as |x| → ∞:

ϕ(x) = −\frac{C_{0}}{|x|^{m−2}} ∫_{ℝ^{m}}(V ϕ)(y)dy

− \frac{1}{ω_{m−1}} ∑_{j=1}^{m} \frac{x_{j}}{|x|^{m}} ∫_{ℝ^{m}} y_{j}(V ϕ)(y)dy + O(|x|^{−m}). (3.8)

(2) For ϕ ∈ N \ E_{0}, ϕ ⊗ ϕ ∉ B(L^{p}(ℝ^{m})) for any 1 ≤ p ≤ ∞ if m = 3 or m = 4 and, if m ≥ 5, ϕ ⊗ ϕ ∈ B(L^{p}(ℝ^{m})) if and only if \frac{m}{m−2} < p < \frac{m}{2}. If ϕ ∈ E_{0} \ E_{1}, then ϕ ⊗ ϕ ∈ B(L^{p}(ℝ^{m})) if and only if \frac{m}{m−1} < p < m for any m ≥ 3 and, if ϕ ∈ E_{1}, then ϕ ⊗ ϕ ∈ B(L^{p}(ℝ^{m})) for all 1 < p < ∞.

(3) If ⟨x⟩^{2}u ∈ L^{1}(ℝ^{m}), |D|^{−1}u(x) has the following expansion as |x| → ∞:

\frac{∫_{ℝ^{m}} udx}{πω_{m−2}|x|^{m−1}} + ∑_{j=1}^{m} \frac{(m − 1)x_{j}}{πω_{m−2}|x|^{m+1}} ∫_{ℝ^{m}} x_{j}udx + O(|x|^{−m−1}). (3.9)

Proof. (1) The smoothness property of ϕ is well known (see e.g. Corollary 2.6 of [2]). We have from (3.7) that

ϕ(x) = −C_{0} ∫_{ℝ^{m}} \frac{V (y)ϕ(y)}{|x − y|^{m−2}} dy.  (3.10)

Taylor’s formula implies that

|\frac{1}{|x − y|^{m−2}} − \frac{1}{|x|^{m−2}} − \frac{(m − 2)x · y}{|x|^{m−1}} | ≤ C \begin{matrix}⟨y⟩ & 2 \\ ⟨x⟩ & m \end{matrix} , |x − y| ≥ 1

and (3.8) follows. Statement (2) follows from (3.8). We omit the proof of (3) which is similar to that of (3.9).

3.1.1 Odd dimensional cases

The structure of singularities depends on m. For odd dimensions m ≥ 3 we have the following results (see, e.g. Theorem 2.12 of [33]). We state it separately for m = 3 and m ≥ 5. In the following Theorems 3.2 and 3.3 for odd m ≥ 3 and Theorem 3.4 for even m ≥ 6, we will indiscriminately write E(λ) for the operator valued function of λ defined near λ = 0 which, when inserted in (3.6) for (1 + G_{0}(λ)V)^{−1}, produces the operator which is bounded in L^{p}(ℝ^{m}) for all 1 ≤ p ≤ ∞.

The case m = 3. By virtue of (3.8), we have for ϕ ∈ N that

ϕ(x) = \frac{L(ϕ)}{|x|} + O(|x|^{−2}) as |x| → ∞, L(ϕ) = \frac{−1}{4π} ∫_{ℝ^{3}} V (x)ϕ(x)dx. (3.11)

Thus, E = {ϕ ∈ N \ {0}: L(ϕ) = 0}(= E_{0}) and, as N ∋ ϕ ↦ L(ϕ) ∈ ℂ is continuous, dim N /E ≤ 1. Any φ ∈ N \ E is called threshold resonance of H. We say that H is of exceptional type of the first kind if E = {0}, the second if E = N and the third kind if {0} ⊊ E ⊊ N. We let D_{0}, D_{1}, … be integral operators defined by

D_{j}u(x) = \frac{1}{4πj!} ∫_{ℝ^{3}} |x − y|^{j−1}u(y)dy, j = 0, 1, ….

so that we have the formal Taylor expansion

G_{0}(λ)u(x) = \frac{1}{4π} ∫_{ℝ^{3}} \frac{e^{iλ|x−y|}}{|x − y|} u(y)dy = ∑_{j=1}^{∞}(iλ)^{j}D_{j}u.

If H is of exceptional type of the third kind, −(V ϕ, ϕ) defines inner product on N and there is a unique real ψ ∈ N such that

−(V ψ, ϕ) = 0, ∀ϕ ∈ E, −(V ψ, ψ) = 1 and L(ψ) > 0.  (3.12)

We define the canonical resonance by

φ = ψ + P V D_{2}V ψ ∈ N .  (3.13)

If H is of exceptional type of the first kind, then dim N = 1 and there is a unique φ ∈ N such that −(V φ, φ) = 1 and L(φ) > 0 and we call this the canonical resonance. We have the following result for m = 3 (see e.g. [33]).

Theorem 3.2. Let m = 3 and let V satisfy |V (x)| ≤ C⟨x⟩^{−δ} for some δ > 3. Suppose that H is of exceptional type of the third kind and let φ be the canonical resonance and a = 4πi|⟨V, φ⟩|^{−2}. Then:

(I + G_{0}(λ)V)^{−1} = \frac{P V}{λ^{2}} + i\frac{P V D_{3}V P V}{λ} − \frac{a}{λ}|φ⟩⟨φ|V + E(λ).  (3.14)

If H is of exceptional type of the first or the second kind, (3.14) holds with P = 0 or φ = 0 respectively.

The case m ≥ 5. If m ≥ 5, (3.8) implies N = E.

Theorem 3.3. Let m ≥ 5 be odd and |V (x)| ≤ C⟨x⟩^{−δ} for some δ > m + 3. Suppose H is of exceptional type. Then:

(1) If m = 5 then, with φ = P V, V being considered as a function,

(I + G_{0}(λ)V)^{−1} = \frac{P V}{λ^{2}} − \frac{a_{0}}{λ} |φ⟩⟨φ|V + E(λ), a_{0} = \frac{i}{24π^{2}} . (3.15)

(2) If m ≥ 7 then

(I + G_{0}(λ)V)^{−1} = \frac{P V}{λ^{2}} + E(λ).  (3.16)

Define S(λ) = (I + G_{0}(λ)V)^{−1} − E(λ) and

Z_{s} = \frac{i}{π} ∫_{0}^{∞} G_{0}(λ)V S(λ)(G_{0}(λ) − G_{0}(−λ))F (λ)λdλ.  (3.17)

Then, it follows from Theorems 3.2 and 3.3 that Z − Z_{s} ∈ B(L^{p}(ℝ^{m})) for all 1 ≤ p ≤ ∞ and we have only to study Z_{s} in what follows.

3.2 Even dimensional case

When m is even, singular terms of (1 + G_{0}(λ)V)^{−1} may contain logarithmic factors. The following is the improvement of Proposition 3.6 of [8]. We let dim E = d and {ϕ_{1}, …, ϕ_{d}} be the real orthonormal basis of E. For making the expression simpler, we state the theorem for V (1 + G_{0}(λ)V)^{−1}.

Theorem 3.4. Let m ≥ 6 be even. Suppose |V (x)| ≤ C⟨x⟩^{−δ} for δ > m + 4 if m = 6 and for δ > m + 3 if m ≥ 8. Let φ = P V with V being considered as a function. Then, we have the following statements for ℑλ ≥ 0 and log λ such that log λ ∈ ℝ for λ > 0:

(1) If m = 6 then, we have that

V (1 + G_{0}(λ)V)^{−1} = \frac{V P V}{λ^{2}} + \frac{ω_{5}}{(2π)^{6}} log λ(V φ ⊗ V φ)

+ (\frac{ω_{5}∥φ∥}{(2π)^{6}})^{2} λ^{2} log^{2} λ(V φ ⊗ V φ) + λ^{2} log λF_{2} + V E(λ), (3.18)

where F_{2} is an operator of rank at most 8 such that

F_{2} = ∑_{a,b=1}^{8} φ_{a} ⊗ ψ_{b}, φ_{a}, ψ_{b} ∈ (L^{1} ∩ L^{∞})(ℝ^{6}).  (3.19)

(2) If m ≥ 8, then we have with a constant c_{m} that

V (1 + G_{0}(λ)V)^{−1} = \frac{V P V}{λ^{2}} + c_{m}(V φ ⊗ V φ)λ^{m−6} log λ + V E(λ). (3.20)

(3) If m ≥ 12, then c_{m}(V φ ⊗ V φ)λ^{m−6} log λ of (3.20) may be included in V E(λ).

Proof. We prove (1) only, using the notation of the proof of subsection 3.2.1 of [8]. A slightly more careful look at the argument there shows that, in spite of Eqn.(3.5) of [8], V (1 + G_{0}(λ)V)^{−1} is actually given by

\frac{V P V}{λ^{2}} + V D_{01} log λ + V D_{21}λ^{2} log λ + V D_{22}λ^{2} log^{2} λ + V E(λ).  (3.21)

Here, with F_{jk} = F_{jk}(0), F_{jk}(λ) being defined by (3.16) of [8], and A(0) = (2π)^{−6}ω_{m−1}(1 ⊗ 1), V D_{01} and V D_{22} are rank 1 operators given by

V D_{01} = V P V F_{01}P V = V P V A(0)V P V = \frac{ω_{m−1}}{(2π)^{6}} (V φ ⊗ V φ),  (3.22)

V D_{22} = V (P V F_{01})^{2}P V = V (P V A(0)V P)^{2}V = \frac{ω_{m−1}^{2}}{(2π)^{12}} ∥φ∥^{2}(V φ ⊗ V φ),

where we have used P V Q = P V and V QP = V P and,

V D_{21} = V P V (F_{21} + F_{00}P V F_{01} + F_{01}P V F_{00})P V  (3.23)
− V X(0)\overline{Q}D_{2}V P V F_{01}P V − V X(0)\overline{Q}A(0)V P V  (3.24)
− V P V F_{01}P V QD_{2}V \overline{Q}X(0) − P V QA(0)V \overline{Q}X(0).  (3.25)

It is obvious that the first line (3.23) is of rank at most 4 and of the form ∑ α_{jk}(V ϕ_{j} ⊗ V ϕ_{k}); four other operators are of rank one and of the form f ⊗ g with f ∈ (L^{1} ∩ L^{∞})(ℝ^{6}). We check this for V X(0)\overline{Q}D_{2}V P V F_{01}P V as a prototype. We have D_{2} = D_{0}^{2} and D_{0}V φ = −φ. Thus, (3.22) implies

V X(0)\overline{Q}D_{2}V P V F_{01}P V = −(2π)^{−6}ω_{m−1}(V X(0)\overline{Q}D_{0}φ) ⊗ (V φ).

Here D_{0}φ ∈ C^{2}(ℝ^{6}) and satisfies D_{0}φ≤_{| · |} C⟨x⟩^{−2} by virtue of Lemma 3.1. Hence, a fortiori D_{0}φ ∈ C_{0}(ℝ^{6}), the Banach space of continuous functions which converge to 0 as |x| → ∞. It is obvious that X ≡ \overline{Q}C_{0}(ℝ^{6}) ⊂ C_{0}(ℝ^{6}) and X(0) = N^{−1}(0) = [\overline{Q}(1 + D_{0}V)\overline{Q}]^{−1} is an isomorphism of X. This is because T = \overline{Q}D_{0}V \overline{Q} is compact both in X = \overline{Q}C_{0}(ℝ^{6}) and Y = \overline{Q}L_{−δ+2}^{2}(ℝ^{6}), X ∩ Y is dense in Y and Ker_{Y}(1 + T) = {0} (see e.g. Lemma 2. 11 of [9]). Thus, V X(0)\overline{Q}D_{0}φ(x)≤_{| · |} C⟨x⟩^{−δ}.

It follows from Theorem 3.4 that Zu = Z_{s}u+Z_{log}u modulo the operator which is bounded in L^{p} for all 1 ≤ p ≤ ∞ and we need study

Z_{se} = \frac{i}{π} ∫_{0}^{∞} G_{0}(λ)V P V (G_{0}(λ) − G_{0}(−λ))F (λ)λ^{−1}dλ,  (3.26)
Z_{log} = ∑_{j,k} \frac{i}{π} ∫_{0}^{∞} G_{0}(λ)λ^{2j}(log λ)^{k}D_{jk}(G_{0}(λ) − G_{0}(−λ))F (λ)λdλ, (3.27)

for even m ≥ 6, where the sum and D_{jk} are as in Theorem 3.4.

4 Proof of Theorem 1.3

The proof of Theorem 1.3 for m = 3 is the simplest and is the prototype for other dimensions and, most of the basic ideas already appear here.

4.1 The case of exceptional type of the first kind

We begin with the case that H is of exceptional type of the first kind and, we let φ be the canonical resonance, a = 4πi|⟨V, φ⟩|^{−2} ≠ 0 and

ψ(x) = |D|^{−1}(V φ)(x) = \frac{1}{2π^{2}} ∫ \frac{V (y)φ(y)}{|x − y|^{2}}dy.  (4.1)

The following lemma proves Theorem 1.3 when H is of exceptional type of the first kind.

408 K. Yajima Lemma 4.1. (1) For 1 < p < 3, there exists a constant C_{p} such that

∥Z_{s}u∥_{p} ≤ C_{p}∥u∥_{p}, u ∈ C_{0}^{∞}(ℝ^{3}).  (4.2)

(2) For 3 < p < ∞, there exists a constant C_{p} such that

∥(Z_{s} + aφ ⊗ ψ)u∥_{p} ≤ C_{p}∥u∥_{p}, u ∈ C_{0}^{∞}(ℝ^{3}).  (4.3)

(3) For p ≥ 3, Z_{s} is unbounded in L^{p}(ℝ^{3}). Proof. Recall c_{0} = C_{0}ω_{2} = 1. We have S(λ) = −\frac{a}{λ}|φ⟩⟨φ|V and

Z_{s}u = −\frac{ia}{π} ∫_{0}^{∞} G_{0}(λ)V φ⟩⟨V φ|(G_{0}(λ) − G_{0}(−λ))u⟩F (λ)dλ.  (4.4)

Defining M(r) = M(r, (V φ) ∗ ǔ), we substitute (2.2) and (2.17) respectively for G_{0}(λ) and ⟨V φ|(G_{0}(λ) − G_{0}(−λ))u⟩. Then,

Z_{s}u = \frac{ai}{π} ∫_{0}^{∞} (∫_{ℝ^{3}} \frac{e^{iλ|x−y|}V (y)φ(y)}{4π|x − y|}dy) (∫_{ℝ} e^{−iλr}rM(r)dr) F (λ)dλ.

If we change the order of integrations,

Z_{s}u = \frac{ai}{2π} ∫_{ℝ^{3}} \frac{K_{0}(|x − y|)V (y)φ(y)}{|x − y|}dy,  (4.5)
K_{0}(ρ) = \frac{1}{2π} ∫_{0}^{∞} e^{iλρ}F (λ) (∫_{ℝ} e^{−irλ}rM(r)dr) dλ.  (4.6)

Since F^{∗}F ∈ S(ℝ), it follows by virtue of Lemmas 2.8 and 2.9 that

K_{0}(ρ) = {(F^{∗}F) ∗ H(rM(r))}(ρ)≤_{| · |} CMH(rM)(ρ).  (4.7)

Function K_{0}(ρ) may also be expressed as

K_{0}(ρ) = \frac{i}{2πρ} ∫_{0}^{∞} e^{iλρ} (F (λ) ∫_{ℝ} e^{−irλ}rM(r)dr)′ dλ.  (4.8)

and, after integration by parts, we see that K_{0}(ρ) satisfies also

K_{0}(ρ)≤_{| · |} Cρ^{−1}(MH(r^{2}M)(ρ) + MH(rM)(ρ)).  (4.9)

The boundary term does not appear in (4.8) since ∫_{ℝ} rM(r)dr = 0. (1a) Let 3/2 < p < 3. By virtue of Young’s inequality

∥Z_{s}u∥_{p} ≤ \frac{|a|(4π)^{1/p}}{2π} ∥V φ∥_{1} (∫_{0}^{∞} |\frac{K_{0}(ρ)}{ρ}|^{p} ρ^{2}dρ)^{1/p} .  (4.10)

Documenta Mathematica 21 (2016) 391–443

We estimate K_{0}(ρ) by (4.7) and use that ρ^{2−p} is an A_{p} weight on ℝ. Lemma 2.7 and Young’s inequality imply

(∫_{0}^{∞} |\frac{K_{0}(ρ)}{ρ}|^{p} ρ^{2}dρ)^{1/p} ≤ C (∫_{0}^{∞} |MH(rM)(ρ)|^{p}ρ^{2−p}dρ)^{1/p}

≤ C_{p} (∫_{0}^{∞} M(r)^{p}r^{2}dr)^{1/p} ≤ C_{p}∥V φ ∗ u∥_{p} ≤ C_{p}∥V φ∥_{1}∥u∥_{p}.  (4.11)

and ∥Z_{s}u∥_{p} ≤ C_{p}∥V φ∥_{1}^{2}∥u∥_{p}. (1b) For 1 < p < \frac{3}{2}, we use estimate (4.9) and that ρ^{2−2p} is an A_{p} weight on ℝ and obtain that

(∫_{0}^{∞} |\frac{K_{0}(ρ)}{ρ}|^{p} ρ^{2}dρ)^{\frac{1}{p}} ≤ (∫_{0}^{∞} |(MH(r^{2}M) + MH(rM))(ρ)|^{p}ρ^{2−2p}dρ)^{\frac{1}{p}}

≤ C (∫_{0}^{∞} |M(r)|^{p} max(r^{2}, r^{2−p})dr)^{\frac{1}{p}} ≤ C(∥V φ∥_{1} + ∥V φ∥_{p′})∥u∥_{p}, (4.12)

where we estimated the integral over 0 ≤ r ≤ 1 by using that

sup |M(r)| ≤ ∥V φ ∗ u∥_{∞} ≤ ∥V φ∥_{p′}∥u∥_{p}.  (4.13)

Thus, we have ∥Z_{s}u∥_{p} ≤ C(∥V φ∥_{1} + ∥V φ∥_{p′})∥V φ∥_{1}∥u∥_{p} for 1 < p < 3/2. Combining (1a) and (1b), we obtain (4.2) for 1 < p < 3 by interpolation([5]).

(2) Let p > 3. Writing ∫_{ℝ} re^{−irλ}M(r)dr = i (∫_{ℝ} e^{−irλ}M(r)dr)′ in (4.6), we apply integration by parts and obtain yet another expression of K_{0}(ρ):

K_{0}(ρ) = \frac{−i}{2π} ∫_{ℝ} M(r)dr − \frac{i}{2π} ∫_{0}^{∞} (e^{iλρ}F (λ))′ (∫_{ℝ} e^{−irλ}M(r)dr) dλ. (4.14)

Denote the second term by K̃_{0}(ρ). By virtue of Lemmas 2.8 and 2.9,

K̃_{0}(ρ)≤_{| · |} C(ρ + 1)MH(M)(ρ).  (4.15)

Substituting (4.14) for K_{0}(ρ) in (4.5), we obtain Z_{s}u = Z_{b}u+Z_{i}u, where Z_{b} and Z_{i} are operators produced by \frac{−i}{2π} ∫_{ℝ} M(r)dr and K̃_{0}(ρ), respectively. Because

\frac{1}{π} ∫_{ℝ} M(r)dr = \frac{1}{2π^{2}} ∫_{ℝ^{3}} (∫_{ℝ^{3}} \frac{(V φ)(x + y)}{|x|^{2}}dx) u(y)dy = ⟨ψ, u⟩  (4.16)

by the definition (4.1), we have by using (3.10) for m = 3 that

Z_{b}u(x) = \frac{a}{4π^{2}} ∫_{ℝ} M(r)dr · ∫_{ℝ^{3}} \frac{V (y)φ(y)}{|x − y|}dy = −a|φ⟩⟨ψ|u⟩.  (4.17)

We splite the integral as

Z_{i}u(x) = \frac{ai}{2π} (∫_{|y|≤1} + ∫_{|y|>1}) \frac{K̃_{0}(|y|)(V φ)(x − y)}{|y|}dy = I_{1}(x)+I_{2}(x). (4.18)

Documenta Mathematica 21 (2016) 391–443

For estimating I_{2} we use (4.15) for ρ ≥ 1: |K̃_{0}(ρ)| ≤ CρMH(M)(ρ). Since ρ^{2} is an A_{p}-weight on ℝ for p > 3, we have by using Young’s and Hölder’s inequalities and Lemma 2.7 that

∥I_{2}∥_{p} ≤ C∥V φ∥_{1} (∫_{0}^{∞} |MH(M)(ρ)|^{p}ρ^{2}dρ)^{\frac{1}{p}}

≤ C∥V φ∥_{1} (∫_{0}^{∞} |M(r)|^{p}r^{2}dr)^{\frac{1}{p}} ≤ C∥V φ∥_{1}^{2}∥u∥_{p}.  (4.19)

Hölder’s inequality implies, with p′ = p/p − 1, that

|I_{1}(x)| ≤ C (∫_{|y|≤1} |\frac{(V φ)(x − y)}{|y|}|^{p′} dy)^{1/p′} (∫_{0}^{1} |K̃_{0}(ρ)|^{p}ρ^{2}dρ)^{1/p} .

Since K̃_{0}(ρ)≤_{| · |} CMH(M)(ρ) for 0 < ρ < 1 by virtue of (4.15) and since ρ^{2} is an A_{p}-weight, we obtain as in (4.19) that

(∫_{0}^{1} |K̃_{0}(ρ)|^{p}ρ^{2}dρ)^{1/p} ≤ C (∫_{0}^{∞} |MH(M)(ρ)|^{p}ρ^{2}dρ)^{1/p} ≤ C∥u∥_{p}. (4.20)

It follows by virtue of Minkowski’s inequality that

∥I_{1}∥_{p} ≤ C∥u∥_{p} ∥(∫_{|y|≤1} |\frac{(V φ)(x − y)}{|y|}|^{p′} dy)^{1/p′} ∥_{p} ≤ C∥u∥_{p}∥V φ∥_{p} (4.21)

because 1 < p′ < 3/2 < 3 < p < ∞. Thus,

∥∫_{ℝ^{3}} \frac{K̃_{0}(|x − y|)V (y)φ(y)}{|x − y|}dy∥_{p} ≤ C(∥V φ∥_{p} + ∥V φ∥_{1})∥u∥_{p}.

With (4.17) this proves (4.3).

(3) Since ∫_{ℝ^{3}} V φdx ≠ 0, Lemma 3.1 implies that φ ∉ L^{p}(ℝ^{3}) for 1 ≤ p ≤ 3 and that ψ ∈ L^{p}(ℝ^{3})^{∗} if and only if p > 3. Hence, φ⊗ψ is unbounded in L^{p}(ℝ^{3}) for any 1 ≤ p ≤ ∞. Thus, statement (2) implies that Z_{s} is unbounded in L^{p}(ℝ^{3}) for p ≥ 3. This completes the proof of the lemma.

We review here the basic strategy of this subsection as it will be repeatedly employed in the following (sub)sections. We express Z_{s}u as the convolution (4.5) of V φ and K_{0}(ρ) of (4.6). By applying integration by parts if necessary we represent and estimate K_{0}(ρ) as in (4.7), (4.9) or (4.15) by using MH. These estimates are used for proving

(∫_{0}^{∞} |K_{0}(ρ)|^{p} ρ^{2−p}dρ)^{\frac{1}{p}} (= ω_{2}^{− \frac{1}{p}} ∥\frac{K_{0}(|x|)}{|x|}∥_{p}) ≤ C∥u∥_{p}  (4.22)

via the weighted inequality for \frac{3}{2} < p < 3, 1 < p < \frac{3}{2} and p > 3 respectively. Desired estimates are then obtained by combining (4.22) and Young’s inequality. However, the boundary term appears in the integration by parts for large values of p > 3 which obstructs the L^{p}-boundedness. We represent the obstruction explicitly in terms of functions of N and show that L^{p}-boundedness depends on the properties of functions in N. Suitable modifications, improvements and additional arguments will be of course necessary in what follows.

4.2 The cases of the second and third kinds

Let H be of exceptional type of the second kind. Then,

S(λ) = \frac{P V}{λ^{2}} + i\frac{P V D_{3}V P V}{λ} ,  (4.23)

where D_{3} is the integral operator with kernel |x − y|^{2}/4π. We take the real orthonormal basis {ϕ_{1}, …, ϕ_{n}} of E and define a_{jk} = π^{−1}⟨ϕ_{j}|V D_{3}V |ϕ_{k}⟩ ∈ ℝ. We have ⟨V, ϕ_{j}⟩ = 0, 1 ≤ j ≤ n. Substituting (4.23) for S(λ) in (3.17), we have

Z_{s}u = Z_{s0}u + Z_{s1}u = ∑_{j,k=1}^{n} Z_{s0,jk}u + ∑_{j=1}^{n} Z_{s1,j},  (4.24)

Z_{s0,jk}u = ia_{jk} ∫_{0}^{∞} G_{0}(λ)V ϕ_{j}⟩⟨V ϕ_{k}|(G_{0}(λ) − G_{0}(−λ))u⟩F (λ)dλ, (4.25)
Z_{s1,j}u = \frac{i}{π} ∫_{0}^{∞} G_{0}(λ)V ϕ_{j}⟩⟨V ϕ_{j}|(G_{0}(λ) − G_{0}(−λ))u⟩F (λ)\frac{dλ}{λ}.  (4.26)

Lemma 4.2. For any 1 < p < ∞, there exists a constant C_{p} such that

∥Z_{s0}u∥_{p} ≤ C_{p}∥u∥_{p}, u ∈ C_{0}^{∞}(ℝ^{3}).  (4.27)

Proof. The operator Z_{s0,jk} is equal to Z_{s} of (4.4) with two φ ∈ N ’s being replaced by ϕ_{j} and ϕ_{k} ∈ E and a by −πa_{jk}. Thus, the proof of Lemma 4.1 implies that Z_{s0,jk} ∈ B(L^{p}(ℝ^{3})) for 1 < p < 3 and that

Z_{s0,jk} − πa_{jk}ϕ_{j} ⊗ |D|^{−1}(V ϕ_{k}) ∈ B(L^{p}(ℝ^{3})), p > 3.  (4.28)

Here ϕ_{j} ⊗|D|^{−1}(V ϕ_{k}) is bounded in L^{p}(ℝ^{3}) for p > 3 because ϕ_{j} ∈ L^{p}(ℝ^{3}) and |D|^{−1}(V ϕ_{k}) ∈ (L^{p}(ℝ^{3}))^{∗} by virtue of (3.8) and (3.9). Thus Z_{s0,jk} ∈ B(L^{p}(ℝ^{3})) for 3 < p and, hence, for 1 < p < ∞ by interpolation. This proves the lemma.

Lemma 4.3. (1) Let 1 < p < 3. Then, for a constant C_{p}, we have

∥Z_{s1}u∥_{p} ≤ C_{p}∥u∥_{p}, u ∈ C_{0}^{∞}(ℝ^{3}).  (4.29)

(2) Let 3 < p < ∞. Then, for a constant C_{p}, we have

∥(Z_{s1} + P)u∥_{p} ≤ C∥u∥_{p}, u ∈ C_{0}^{∞}(ℝ^{3}).  (4.30)

In (4.30) P may be replaced by P ⊖ P_{1} by virtue of Lemma 3.1.

(3) The operator Z_{s1} is bounded in L^{p}(ℝ^{3}) for some p > 3 if and only if E = E_{1}. In this case Z_{s1} is bounded in L^{p}(ℝ^{3}) for all 1 < p < ∞.

Proof. Define ψ_{j}(x) = |D|^{−1}(V ϕ_{j})(x), j = 1, …, n. Then Lemma 2.3 implies

Z_{s1,j}u = \frac{i}{π} ∫_{0}^{∞} G_{0}(λ)|V ϕ_{j}⟩⟨ψ_{j}|(G_{0}(λ) − G_{0}(−λ))u⟩F (λ)dλ  (4.31)

which can be obtained from Z_{s}u of (4.4) by replacing a by −1, the first V φ by V ϕ_{j} and the second by ψ_{j}. Thus, it may be expressed by using K_{0,j}(ρ) of (4.6) with M(r) being replaced by M_{j}(r) = M(r, ψ_{j} ∗ ǔ):

Z_{s1,j}u = \frac{1}{2πi} ∫_{ℝ^{3}} \frac{K_{0,j}(|x − y|)V (y)ϕ_{j}(y)}{|x − y|}dy.  (4.32)

(1) The argument of (1a) in the proof of Lemma 4.1 implies

∥Z_{s1,j}u∥_{p} ≤ C∥V ϕ_{j}∥_{1}∥ψ_{j} ∗ u∥_{p}, 3/2 < p < 3  (4.33)

(see (4.11)) and the one of (1b) does

∥Z_{s1,j}u∥_{p} ≤ C∥V ϕ_{j}∥_{1}(∥ψ_{j} ∗ u∥_{p} + ∥ψ_{j} ∗ u∥_{∞}), 1 < p < 3/2  (4.34)

(see (4.12)). Since ∫ V ϕ_{j}dx = 0, (3.9) implies that ψ_{j} = |D|^{−1}ϕ_{j} ∈ L^{q}(ℝ^{3}) for all 1 < q ≤ ∞ and that the convolution operator with ψ_{j}(x) is bounded in L^{p} for any 1 < p < ∞ via Calderón-Zygmund theory (see e.g. [26], pp. 30-36). Thus, ∥ψ_{j} ∗ u∥_{p} ≤ C∥u∥_{p}, ∥ψ_{j} ∗ u∥_{∞} ≤ ∥ψ_{j}∥_{p′}∥u∥_{p} and Z_{s1,j} is bounded in L^{p}(ℝ^{3}) for all 1 < p < 3, j = 1, …, n. Statement (1) follows.

(2) Integration by parts as in (4.14) by using the identity ∫_{ℝ} e^{−irλ}rM_{j}(r)dr = i (∫_{ℝ} e^{−irλ}M_{j}(r)dr)′ implies that K_{0,j}(ρ) may be written as

−\frac{i}{2π} ∫_{ℝ} M_{j}(r)dr − \frac{i}{2π} ∫_{0}^{∞} (e^{iλρ}F (λ))′ (∫_{ℝ} e^{−irλ}M_{j}(r)dr) dλ,  (4.35)

which we insert into (4.32). Since |D|^{−1}ψ_{j} = (−Δ)^{−1}(V ϕ_{j}) = −ϕ_{j}, (4.16) with ψ_{j} ∈ E in place of V φ produces −⟨ϕ_{j}|u⟩. It follows that the boundary term of (4.35) produces

\frac{−1}{4π} ∫_{ℝ^{3}} \frac{V (y)ϕ_{j}(y)}{|x − y|}dy · \frac{1}{π} ∫_{ℝ} M_{j}(r)dr = −|ϕ_{j}⟩⟨ϕ_{j}|u⟩  (4.36)

as in (4.17). Denote by K̃_{0j}(ρ) and Z̃_{s1,j} the second term of (4.35) and the operator it produces via (4.32). They can respectively be obtained from K̃_{0}(ρ) of (4.14) and Z_{i} of (4.18) by replacing M(r) and K̃_{0}(ρ) by M_{j}(r) and K̃_{0,j}(ρ). Thus, the argument of step (2) of the proof of Lemma 4.1, (4.19) and (4.21) in particular, implies that

∥Z̃_{s1,j}u∥_{p} ≤ C(∥V ϕ_{j}∥_{p} + ∥V ϕ_{j}∥_{1})∥ψ_{j} ∗ u∥_{p}, 3 < p < ∞.  (4.37)

The Calderón-Zygmund theory with (3.9) once more implies ∥Z̃_{s1,j}u∥_{p} ≤

C∥u∥_{p}. Since ϕ ⊗ ϕ ∈ B(L^{p}) for all 1 < p < ∞ if ϕ ∈ E_{1} by virtue of (3.8), this together with (4.36) proves statement (2).

(3) It is obvious from (1) and (2) that Z_{s1} ∈ B(L^{p}(ℝ^{3})) for all 1 < p < ∞ if E = E_{1}. Suppose then that Z_{s1} ∈ B(L^{p}(ℝ^{3})) for some p > 3 then P ⊖ P_{1} must be bounded in L^{p}(ℝ^{3}) by virtue of (2). Take the orthonormal basis {ϕ_{1}, …, ϕ_{d}} of E ⊖ E_{1} and {ρ_{1}, …, ρ_{d}} ⊂ C_{0}^{∞}(ℝ^{3}) such that {(ρ_{j}, ϕ_{k})} becomes the unit matrix. Then, (P ⊖ P_{1})ρ_{j} = ϕ_{j}, j = 1, …, n and, if P ⊖ P_{1} is bounded in L^{p}(ℝ^{3}) for some p ≥ 3, there must exist a constant C > 0 such that

|(u, ϕ_{j})| = |((P ⊖ P_{1})u, ρ_{j})| ≤ C_{j}∥u∥_{p}, for all u ∈ C_{0}^{∞}(ℝ^{3}).

Then, ϕ_{j} has to be in L^{p′}(ℝ^{3}) for p′ ≤ 3/2 for all j = 1, …, n. This implies ϕ_{j} = 0 by virtue of (3.8). Thus, E = E_{1} must hold. This completes the proof.

Lemma 4.2 and Lemma 4.3 prove Theorem 1.3 when H is of exceptional type of the second kind. The following lemma completes the proof of Theorem 1.3.

Lemma 4.4. Suppose that H is of exceptional type of the third kind. Then:

(1) W is bounded in L^{p}(ℝ^{3}) for all 1 < p < 3.

(2) W + aφ ⊗ (|D|^{−1}V φ) + P is bounded in L^{p}(ℝ^{3}) for all p > 3.

(3) W is unbounded in L^{p}(ℝ^{3}) for any p > 3 and p = 1.

Proof. The combination of Lemmas 4.1, 4.2 and 4.3 proves statements (1) and (2). Suppose that W is bounded in L^{p}(ℝ^{3}) for some 3 < p < ∞. Then, so is a(φ ⊗ (|D|^{−1}V φ)) + P. Let ψ ∈ N be the function which defines the canonical resonance φ by (3.13) and which satisfies (3.12). Then,

(V ψ, a(φ ⊗ (|D|^{−1}V φ))u + P u) = −a(|D|^{−1}V φ, u), u ∈ C_{0}^{∞}(ℝ^{3})

and this must be extended to a bounded functional of u ∈ L^{p}(ℝ^{3}). Hence, |D|^{−1}V φ ∈ L^{q}(ℝ^{3}) for q = (p − 1)/p < 3/2. This contradicts (3.8) because ∫_{ℝ^{3}} V (x)φ(x)dx ≠ 0 and (3) is proved.

5 Proof of Theorems 1.4 and 1.5 for odd m

If m ≥ 5, then N = E and we let {ϕ_{1}, …, ϕ_{d}} be the real orthonormal basis of E. Theorem 3.3 implies that, with a_{0} = i/(24π^{2}),

S(λ) = \{ \begin{matrix}λ^{−2}P V − a_{0}λ^{−1}(φ ⊗ V φ), & if m = 5, \\ λ^{−2}P V, & if m ≥ 7. \end{matrix}  (5.1)

Note that φ ≠ 0 if and only if E_{1} ≠ E. We substitute (5.1) for S(λ) in (3.17) and apply (2.2) and (2.17) as previously. Let C_{j}, c_{k}, 1 ≤ j, k ≤ \frac{m−3}{2} respectively be constants of (2.2) and (2.17). Then, we have

Z_{s}u = Z_{s0}u + Z_{s1}u,  (5.2)

where Z_{s0} = 0 for m ≥ 7 and, for m = 5, with M(r) = M(r, V φ ∗ ǔ)

Z_{s0}u = −2ia_{0} ∑_{j,k=0,1}(−1)^{j+1}C_{k}c_{j}Z_{s0}^{jk}u,  (5.3)

Z_{s0}^{jk}u(x) = ∫_{ℝ^{5}} \frac{V φ(y)}{|x − y|^{3−k}} K_{0}^{(j,k)}(|x − y|)dy,  (5.4)
K_{0}^{(j,k)}(ρ) = \frac{1}{2π} ∫_{0}^{∞} e^{iλρ}λ^{j+k} (∫_{ℝ} e^{−iλr}r^{j+1}M(r)dr) F (λ)dλ,  (5.5)

and Z_{s1}u is defined for all m ≥ 5 by

Z_{s1}u = ∑_{l=1}^{d} Z_{s1}(ϕ_{l})u  (5.6)

where, for ϕ ∈ E, with M(r) = M(r, V ϕ ∗ ǔ),

Z_{s1}(ϕ)u = 2i ∑_{j,k=0}^{\frac{m−3}{2}}(−1)^{j+1}C_{k}c_{j}Z_{s1}^{jk}(ϕ),  (5.7)

Z_{s1}^{jk}(ϕ)u(x) = ∫_{ℝ^{m}} \frac{V ϕ(y)}{|x − y|^{m−2−k}} K^{(j,k)}(|x − y|)dy,  (5.8)
K^{(j,k)}(ρ) = \frac{1}{2π} ∫_{0}^{∞} e^{iλρ}λ^{j+k−1} (∫_{ℝ} e^{−iλr}r^{j+1}M(r)dr) F (λ)dλ.  (5.9)

Note that Z_{s0}^{jk}u and K_{0}^{(j,k)}(ρ) are obtained from Z_{s1}^{jk}u and K^{(j,k)}(ρ) by changing ϕ by φ and λ^{j+k−1} by λ^{j+k} in (5.9). We shall prove the last statements of (2) and (3) of Theorems 1.4 and 1.5 only for Z_{s1}(ϕ) since the proof of Lemma 4.3 (3) can easily be adapted for proving the same statements for Z_{s0}.

5.1 Estimate of Z_{s0} for m = 5

We begin by proving the following lemma for Z_{s0}, assuming φ ≠ 0.

Lemma 5.1. (1) Z_{s0} is bounded in L^{p}(ℝ^{5}) for 1 < p < 5.

(2) Z_{s0} + a_{0}|φ⟩⟨|D|^{−1}(V φ)| is bounded in L^{p}(ℝ^{5}) for 5/2 < p < ∞.

(3) Z_{s0} is not bounded in L^{p}(ℝ^{5}) if p ≥ 5.

Proof. For φ = P V, we have ∫_{ℝ^{5}} V φdx = ∥φ∥^{2} > 0 and, by virtue of (3.8) and (3.9), φ ⊗ |D|^{−1}(V φ) ∈ B(L^{p}(ℝ^{5})) if and only if 5/3 < p < 5. Hence, statement (3) follows (2). Using that e^{iρλ} = (iρ)^{−(k+1)}∂_{λ}^{k+1}e^{iρλ} and ∫_{ℝ} λ^{j+1}e^{−iλr}M(r)dr = i^{j} (∫_{ℝ} e^{−iλr}M(r)dr)^{(j)}, we apply integration by parts to (5.5) and write K_{0}^{(j,k)}(ρ) in two ways

K_{0}^{(j,k)}(ρ) = \frac{i^{k+1}}{2πρ^{k+1}} ∫_{0}^{∞} e^{iρλ} (λ^{j+k}F (λ) ∫_{ℝ} e^{−iλr}r^{j+1}M(r)dr)^{(k+1)} dλ (5.10)

= \frac{(−i)^{j}}{2π} ∫_{0}^{∞} (e^{iλρ}λ^{j+k}F (λ))^{(j)} (∫_{ℝ} e^{−iλr}rM(r)dr) dλ. (5.11)

Note that boundary terms do not appear in (5.10) since ∫_{ℝ} rM(r)dr = 0 and, if k = 1, we may apply further integration by parts to (5.11) without having boundary term and

K_{0}^{(j,k)}(ρ) = \frac{(−i)^{j+1}}{2π} ∫_{0}^{∞} (e^{iλρ}λ^{j+k}F (λ))^{(j+1)} (∫_{ℝ} e^{−iλr}M(r)dr) dλ. (5.12)

We then apply Lemmas 2.8 and 2.9 to the right sides and obtain the following estimates for j, k = 0, 1:

K_{0}^{(j,k)}(ρ)≤_{| · |} \{ \begin{matrix}Cρ^{−(k+1)} ∑_{l=0}^{k+1} MH(r^{j+l+1}M)(ρ), & (5.13) \\ C(1 + ρ^{j+k})MH(r^{1−k}M)(ρ). & (5.14) \end{matrix}

(a) Let 1 < p < 5/4. Since |r|^{−4(p−1)} is an A_{p} weight on ℝ and 3p − 4 > −1, we have by using (5.13) and (4.13) that, for any j, k = 0, 1,

∥\frac{K_{0}^{(j,k)}(|y|)}{|y|^{3−k}} ∥_{p} ≤ C ∑_{l=0}^{k+1} (∫_{0}^{∞} \frac{|MH(r^{j+l+1}M)(ρ)|^{p}}{ρ^{4(p−1)}} dρ)^{1/p}

≤ C (∫_{0}^{1} \frac{|M(r)|^{p}dr}{r^{3p−4}} + ∫_{1}^{∞} |M(r)|^{p}r^{4}dr)^{\frac{1}{p}} ≤ C(∥V φ∥_{p′} + ∥V φ∥_{1})∥u∥_{p}. (5.15)

Young’s inequality then implies ∥Z_{s0}^{jk}u∥_{p} ≤ C∥V φ∥_{1}(∥V φ∥_{p′} + ∥V φ∥_{1})∥u∥_{p}.

(b) We next show that ∥Z_{s0}^{j1}u∥_{p} ≤ C∥u∥_{p} for p > 5 and j = 0, 1. Interpolating this with the result of (a), we then have the same for all 1 < p < ∞. We split the integral as in (4.18) and repeat the argument after it:

|Z_{0s}^{j1}u(x)| ≤ C (∫_{|y|≤1} + ∫_{|y|>1}) \frac{|V φ(x − y)|}{|y|^{2}}|K_{0}^{(j,1)}(|y|)|dy = I_{1}(x) + I_{2}(x).

For ρ ≥ 1, we have K_{0}^{(j,1)}(ρ)≤_{| · |} Cρ^{2}MH(M(r))(ρ) by virtue of (5.14) and since r^{4} is A_{p} weight on ℝ if p > 5. It follows that

∥I_{2}∥_{p} ≤ C∥V φ∥_{1} ∥\frac{K_{0}^{(j,1)}}{|x|^{2}} ∥_{L^{p}(|x|≥1)} ≤ C∥V φ∥_{1} (∫_{0}^{∞} |MH(M)(ρ)|^{p}ρ^{4}dρ)^{\frac{1}{p}}

≤ C∥V φ∥_{1} (∫_{0}^{∞} |M(r)|^{p}r^{4}dr)^{1/p} ≤ C∥V φ∥_{1}^{2}∥u∥_{p}.  (5.16)

Hölder’s inequality and (5.14) for 0 ≤ ρ ≤ 1, K_{0}^{(j,1)}(ρ)≤_{| · |} CMH(M)(ρ), imply

|I_{1}(x)| ≤ C (∫_{|y|≤1} |\frac{|V φ(x − y)|}{|y|^{2}} |^{p′} dy)^{\frac{1}{p′}} (∫_{0}^{1} |MH(M(r))(ρ)|^{p}ρ^{4}dρ)^{\frac{1}{p}} .

Since p′ ≤ \frac{5}{4} if p > 5, Minkowski’s inequality and (5.16) imply

∥I_{1}∥_{p} ≤ C∥V φ∥_{1}∥V φ∥_{p}∥u∥_{p}.  (5.17)

(c) We finally prove −2ia_{0}C_{0}(c_{1}Z_{s0}^{10} − c_{0}Z_{s0}^{00}) + a_{0}|φ⟩⟨|D|^{−1}(V φ)| ∈ B(L^{p}(ℝ^{5})) for p > 5/2. This will complete the proof of the lemma because this and (b) imply statement (2) by virtue of (5.3); since |φ⟩⟨|D|^{−1}(V φ)| is bounded in L^{p}(ℝ^{5}) for 5/3 < p < 5 as remarked previously, this also implies −2ia_{0}C_{0}(c_{1}Z_{s0}^{10}−c_{0}Z_{s0}^{00}) ∈ B(L^{p}(ℝ^{5})) for 5/3 < p < 5 and, hence, for 1 < p < 5 by virtue of result (a) and interpolation. Then, (b) yields statement (1). If k = 0, further integration by parts to (5.11) produces boundary term:

K_{0}^{(j,0)}(ρ) = \frac{(−i)^{j+1}}{2π} j! ∫_{ℝ} M(r)dr

+ \frac{(−i)^{j+1}}{2π} ∫_{0}^{∞} (e^{iλρ}λ^{j}F (λ))^{(j+1)} (∫_{ℝ} e^{−iλr}M(r)dr) dλ. (5.18)

The second integral, which we denote by K̃_{0}^{(j,0)}(ρ), satisfies

K̃_{0}^{(j,0)}(ρ)≤_{| · |} C(1 + ρ^{j+1})MH(M)(ρ) ≤ C(1 + ρ^{j+2})MH(M)(ρ) (5.19)

and we estimate the operator Z̃^{j0} obtained by replacing K_{0}^{(j,0)}(ρ) by K̃_{0}^{(j,0)}(ρ) in (5.4) by repeating the argument of step (b): Split Z̃^{j0}u(x) as in step (b) and obtain ∥I_{2}∥_{p} ≤ C∥u∥_{p} for 5/2 < p < 5 (resp. p > 5) by using the first (resp. second) estimate of (5.19) and that r^{4−p} (resp. r^{4}) is an A_{p}- weight on ℝ. Likewise we obtain ∥I_{1}∥_{p} ≤ C∥u∥_{p} for 5/2 < p < 5 (resp. p > 5) by first applying Hölder’s inequality by considering the integrand as (|V φ(x − y)|/|y|^{2}) · (|K̃_{0}^{(j,0)}(|y|)|/|y|) (resp. |V φ(x − y)|/|y|^{3} · |K̃_{0}^{(j,0)}(|y|)|) and then using Minkowski’s inequality. Thus, we have for j = 0, 1 that

∥Z̃^{j0}u∥_{p} ≤ C∥u∥_{p}, 5/2 < p < ∞.  (5.20)

The contribution of boundary terms of (5.18) to c_{0}K_{0}^{(00)} − c_{1}K_{0}^{(10)} is given by virtue of (2.3) and (3.9) by

(c_{1} − ic_{0}) × \frac{1}{2π} ∫_{ℝ} M(r)dr = \frac{c_{0}}{πi} ∫_{ℝ} M(r)dr = −4π^{2}C_{0}i⟨|D|^{−1}(V φ), u⟩

and this contributes to 2a_{0}iC_{0}(c_{0}Z_{s0}^{00} − c_{1}Z_{s0}^{10})u(x) by

8π^{2}a_{0}C_{0}^{2} ∫_{ℝ^{5}} \frac{V φ(y)}{|x − y|^{3}} dy · (⟨|D|^{−1}(V φ), u⟩) = −a_{0}φ(x)⟨|D|^{−1}(V φ), u⟩,

where we used 8π^{2}C_{0} = 1 when m = 5. This proves the lemma.

5.2 Estimates of Z_{s1} for m ≥ 5.

We next study Z_{s1}u for all m ≥ 7. By virtue of (5.6) and (5.7) and the remark at the beginning of section 5, it suffices to study Z_{1s}^{jk}(ϕ)u defined by (5.8) for ϕ ∈ E. For simplifying notation, we often omit ϕ from Z_{1s}^{jk}(ϕ). Define

M_{∗}(r) = M(r, |D|^{−1}(V ϕ) ∗ ǔ).  (5.21)

Then, by virtue of (2.9), K^{(j,k)}(ρ) may also be expressed as

K^{(j,k)}(ρ) = \frac{1}{2π} ∫_{0}^{∞} e^{iλρ}λ^{j+k}F (λ) (∫_{ℝ} e^{−iλr}r^{j+1}M_{∗}(r)dr) dλ  (5.22)

which has the larger factor λ^{k+j} than λ^{k+j−1} of (5.9). We omit the proof of the following lemma which is essentially the same as that of (5.13, 5.14)

Lemma 5.2. K^{(j,k)}(ρ) satisfies the following estimates:

K^{(j,k)}(ρ)≤_{| · |} \{ \begin{matrix}Cρ^{−k−1} ∑_{l=0}^{k+1} MH(r^{j+1+l}M)(ρ), & j ≥ 2. & (5.23) \\ C(1 + ρ^{j−1})MH(r^{2}M)(ρ), & j ≥ 1. & (5.24) \\ C(1 + ρ^{j})MH(rM)(ρ), & k + j ≥ 1. & (5.25) \\ C(1 + ρ^{j+1})MH(M)(ρ), & k ≥ 2. & (5.26) \\ C(1 + ρ^{j})MH(rM_{∗})(ρ), & k ≥ 0. & (5.27) \end{matrix}

Lemma 5.3. Suppose m ≥ 5 and ϕ ∈ E. Then:

(1) If j ≥ 2, Z_{1s}^{jk}(ϕ), k = 0, …, \frac{m−3}{2}, are bounded in L^{p}(ℝ^{m}) for 1 < p < \frac{m}{2}.

(2) For k ≥ 2, Z_{1s}^{jk}(ϕ), j = 0, …, \frac{m−3}{2}, are bounded in L^{p}(ℝ^{m}) for \frac{m}{3} < p.

(3) For all j and k, Z_{1s}^{jk}(ϕ) is bounded in L^{p}(ℝ^{m}) for \frac{m}{3} < p < \frac{m}{2}.

If both j, k ≥ 2, Z_{1s}^{jk}(ϕ) is bounded in L^{p}(ℝ^{m}) for all 1 < p < ∞.

Proof. (a) We first prove (1) for 1 < p < \frac{m}{m−1}. General case follows from this and (3) by interpolation. We use (5.23) and that r^{−(m−1)(p−1)} is an A_{p} weight on ℝ for 1 < p < \frac{m}{m−1}. Then, estimating as in (5.15), we obtain

∥Z_{s1}^{jk}u∥_{p} ≤ C∥V ϕ∥_{1} (∫_{0}^{∞} |M(r)|^{p}r^{m−1}dr + ∫_{0}^{1} \frac{|M(r)|^{p}}{r^{(m−4)p}} r^{m−1}dr)^{1/p}

≤ C∥V ϕ∥_{1}(∥V ϕ∥_{1} + ∥V ϕ∥_{p′})∥u∥_{p}.  (5.28)

(b) We next prove (2) for p > m. General case then follows from this and (3) by interpolation. We split the integral as in (4.18):

Z_{s1}^{jk}u(x)≤_{| · |} (∫_{|y|≤1} + ∫_{|y|≥1}) \frac{|V ϕ(x − y)|}{|y|^{m−2−k}}|K^{(j,k)}(|y|)|dy = I_{1}(x) + I_{2}(x).

Using (5.26) for ρ ≥ 1 and that r^{m−1} is A_{p} weight on ℝ if p > m, we obtain

∥I_{2}∥_{p} ≤ C∥V ϕ∥_{1} (∫_{1}^{∞} |MH(M)(ρ)|^{p}ρ^{m−1}dρ)^{\frac{1}{p}} ≤ C∥V ϕ∥_{1}^{2}∥u∥_{p}. (5.29)

Hölder’s inequality and (5.26) for 0 ≤ ρ ≤ 1 imply that

|I_{1}(x)| ≤ (∫_{|y|≤1} |\frac{V ϕ(x − y)|}{|y|^{m−2−k}} |^{p′} dy)^{1/p′} (∫_{0}^{1} |MH(M)(ρ)|^{p}ρ^{m−1}dρ)^{1/p} .

(5.30)

Then, Minkowski’s inequality and the estimate as in (5.29) yield

∥I_{1}∥_{p} ≤ C∥V ϕ∥_{1}∥u∥_{p} (∫_{|x|<1} \frac{∥V ϕ∥_{p}^{p′}dx}{|x|^{(m−2−k)p′}} )^{1/p′} ≤ C∥V ϕ∥_{1}∥V ϕ∥_{p}∥u∥_{p}

because p′ ≤ \frac{m}{m−1} if p > m and |y|^{−(m−2−k)p′} is integrable over |y| ≤ 1. Thus, statement (2) for p > m follows.

(c) We prove statement (3) by modifying the argument in step (b). Let \frac{m}{3} < p < \frac{m}{2}. Then, r^{m−1−2p} is an A_{p} weight on ℝ. We split the integral of Z_{s1}^{jk}u(x) as in step (b).

(i) Let j ≥ 1. Estimate (5.24) for ρ ≥ 1 and Lemma 2.7 yield

∥I_{2}∥_{p} ≤ C∥V ϕ∥_{1} (∫_{0}^{∞} |MH(r^{2}M)(ρ)|^{p}ρ^{m−1−2p}dρ)^{\frac{1}{p}} ≤ C∥V ϕ∥_{1}^{2}∥u∥_{p}.

(5.31)

Estimate (5.24) for ρ ≤ 1 and Hölder’s inequality imply

|I_{1}(x)| ≤ (∫_{|y|≤1} |\frac{|V ϕ(x − y)|}{|y|^{m−4−k}} |^{p′} dy)^{\frac{1}{p′}} (∫_{0}^{1} |MH(r^{2}M)(ρ)|^{p}ρ^{m−1−2p}dρ)^{\frac{1}{p}} .

Minkowski’s inequality and the second estimate of (5.31) imply ∥I_{1}∥_{p} ≤ C∥V ϕ∥_{p}∥V ϕ∥_{1}∥u∥_{p} as previously and, hence, ∥Z_{s1}^{jk}u∥_{p} ≤ C∥u∥_{p}.

(b) Let j = 0. Express K^{(0,k)}(ρ) by using M̃(r) of (2.15) and estimate as

K^{(0,k)}(ρ) = \frac{1}{2iπ} ∫_{0}^{∞} e^{iλρ}λ^{k} (∫_{ℝ} e^{−iλr}M̃(r)dr) F (λ)dλ≤_{| · |} CMH(M̃)(ρ).

(5.32)

Since ρ^{−(m−2−k)} ≤ ρ^{−2} for ρ ≥ 1, Young’s inequality, Lemma 2.7 and Hardy’s inequality yield

∥I_{2}∥_{p} ≤ C∥V ϕ∥_{1} (∫_{0}^{∞} |M̃(r)|^{p}r^{m−1−2p}dr)^{1/p}

≤ C∥V ϕ∥_{1} (∫_{0}^{∞} |M(r)|^{p}r^{m−1}dr)^{1/p} ≤ C∥V ϕ∥_{1}∥V ϕ∥_{p}∥u∥_{p}.  (5.33)

Hölder’s inequality and (5.32) imply

|I_{1}(x)| ≤ (∫_{|y|≤1} |\frac{|V ϕ(x − y)|}{|y|^{m−4−k}} |^{p′} dy)^{1/p′} (∫_{0}^{1} |MH(M̃)(ρ)|^{p}ρ^{m−1−2p}dρ)^{1/p}

Estimate the second factor by (5.33) and use Minkowski’s equality. This yields ∥I_{1}∥_{p} ≤ C∥V ϕ∥_{p}∥V ϕ∥_{1}∥u∥_{p}. The last statement follows from (1) and (2) by interpolation.

Lemma 5.4. Let m ≥ 5 and ϕ ∈ E. Then:

(1) For 1 < p < \frac{m}{2}, ∥(c_{0}Z_{s1}^{(0,k)} − c_{1}Z_{s1}^{(1,k)})u∥_{p} ≤ C∥u∥_{p} for all 0 ≤ k ≤ \frac{m−3}{2}.

(2) The operator Z_{s1}(ϕ) is bounded in L^{p}(ℝ^{m}) for 1 < p < \frac{m}{2}.

Proof. It suffices to prove the estimate of (1) for 1 < p < \frac{m}{m−1} since that for 1 < p < \frac{m}{2} follows from this and Lemma 5.3 (3) by interpolation and since statement (2) follows from this and statement (1) of Lemma 5.3. Using the identity e^{iλρ} = (iρ)^{−k−1}∂_{λ}^{k+1}e^{iλρ}, we apply integration by parts k + 1 times to the integral of (5.32) and use the identity (2.16). We obtain

K^{(0,k)}(ρ) = \frac{i^{k}}{2πρ^{k+1}} (k! ∫_{ℝ} r^{2}M(r)dr

+ ∑_{l=0}^{k+1} (\begin{matrix}k + 1 \\ l \end{matrix}) ∫_{0}^{∞} e^{iλρ}(λ^{k}F)^{(k+1−l)} ∫_{ℝ} e^{−iλr}(−ir)^{l}M̃drdλ) . (5.34)

Integration by parts k + 1 times to K^{(1,k)}(ρ) of (5.9) likewise yields

K^{(1,k)}(ρ) = \frac{i^{k}}{2πiρ^{k+1}} (−k! ∫_{ℝ} r^{2}M(r)dr

− ∑_{l=0}^{k+1} (\begin{matrix}k + 1 \\ l \end{matrix}) ∫_{0}^{∞} e^{iλρ}(λ^{k}F)^{(k+1−l)} ∫_{ℝ} e^{−iλr}(−ir)^{l}r^{2}Mdrdλ) . (5.35)

Since c_{0} − ic_{1} = 0, the boundary terms of (5.34) and (5.35) cancel out and

\frac{c_{0}K^{(0,k)}(ρ) − c_{1}K^{(1,k)}(ρ)}{ρ^{m−2−k}}≤_{| · |} \frac{C}{ρ^{m−1}} ∑_{l=0}^{k+1}(MH(r^{l}M̃)(ρ) + MH(r^{l+2}M)(ρ)).

For 1 < p < \frac{m}{m−1}, ρ^{−(m−1)(p−1)} is an A_{p}-weight on ℝ. It follows by Young’s inequality, Lemma 2.7 and Hardy’s inequality that ∥(c_{0}Z^{(0,k)} − c_{1}Z^{(1,k)})u∥_{p} is bounded by C∥V ϕ∥_{1} times

∑_{l=0}^{k+1} (∫_{0}^{∞}(|M̃(r)|^{p}r^{pl} + |M(r)|^{p}r^{p(l+2)})r^{m−1−p(m−1)}dr)^{1/p}  (5.36)

≤ C (∫_{0}^{1} \frac{|M(r)|^{p}}{r^{p(m−3)}} r^{m−1}dr + ∫_{0}^{∞} |M(r)|^{p}r^{m−1}dr)^{1/p}  (5.37)
≤ C(∥V ϕ∥_{p′} + ∥V ϕ∥_{p})∥u∥_{p}.  (5.38)

Here we used k + 3 ≤ m − 1 for m ≥ 5 in the first step and p(m − 1) < m in the last. This proves the estimate of (1) for 1 < p < \frac{m}{m−1}.

Lemma 5.1 and the second statement of Lemma 5.4 prove statement (1) of Theorems 1.4 and 1.5 for odd m. The following lemma (and Lemma 5.1 for the case m = 5) proves statement (2) of these theorems for odd m.

Lemma 5.5. Let m ≥ 5, ϕ ∈ E and \frac{m}{2} < p < m. Then, for a constant C > 0,

∥Z_{s1}(ϕ)u + \frac{Γ (\frac{m−2}{2})}{\sqrt{π}Γ (\frac{m−1}{2})}⟨u, ϕ⟩ϕ∥_{p} ≤ C∥u∥_{p}.  (5.39)

If Z_{s1}(ϕ) ∈ B(L^{p}) for some \frac{m}{2} < p < m, then ϕ ∈ E_{0} and Z_{s1}(ϕ) ∈ B(L^{p}) for all 1 < p < m.

Proof. Let j + k ≥ 1. Since m − 2 − (k + j) ≥ 1, we have from (5.25) that

\frac{K^{(j,k)}(ρ)}{ρ^{m−2−k}}≤_{| · |} C (\frac{1}{ρ^{m−2−k}} + \frac{1}{ρ}) MH(rM)(ρ).

Using that r^{m−1−p} is A_{p} weight and (m−2)p′ < m for m/2 < p < m, we repeat the argument of the step (b) or (c) of the proof of Lemma 5.3 and obtain

∥Z_{s1}^{jk}u∥_{p} ≤ C∥u∥_{p}, j + k ≥ 1.  (5.40)

It remains to consider −2iC_{0}c_{0}Z_{s1}^{00}, see (5.7). We apply integration by parts to the right of (5.22) with j = k = 0:

K^{(0,0)}(ρ) = \frac{i}{2π} ∫_{0}^{∞} e^{iλρ}F (λ)∂_{λ} (∫_{ℝ} e^{−iλr}M_{∗}(r)dr) dλ

= \frac{−i}{2π} ∫_{ℝ} M_{∗}(r)dr − \frac{i}{2π} ∫_{0}^{∞}(e^{iλρ}F (λ))′ (∫_{ℝ} e^{−iλr}M_{∗}(r)dr) dλ. (5.41)

We denote the second integral of (5.41) by K_{∗}^{(0,0)}(ρ) and, by Z_{∗}^{00} the operator produced by substituting K_{∗}^{(0,0)}(ρ) for K^{(0,0)}(ρ) in (5.8). We have |K_{∗}^{(0,0)}(ρ)| ≤ C(1 + ρ)MH(M_{∗})(ρ). Decompose

Z_{∗}^{00}u(x)≤_{| · |} (∫_{|y|≤1} + ∫_{|y|≥1}) |(V ϕ)(x − y)|\frac{|K_{∗}^{(0,0)}(|y|)|}{|y|^{m−2}}dy = I_{1}(x) + I_{2}(x)

as previously. For estimating ∥I_{2}∥_{p}, define 1/q = 1/p−1/m and apply Young’s inequality, Hölder’s inequality, Lemma 2.7 noticing that q > m and r^{m−1} is A_{q} weight and, Hardy-Littlewood-Soblev inequality recalling that |D|^{−1}(V ϕ) ∗ (x)≤_{| · |} C⟨x⟩^{1−m}. We obtain

∥I_{2}∥_{p} ≤ C∥V ϕ∥_{1} (∫_{0}^{∞} |MH(M_{∗})(ρ)|^{q}ρ^{m−1}dρ)^{1/q} ∥\frac{1}{|y|^{m−3}} ∥_{L^{m}(|y|>1)}

≤ C∥V ϕ∥_{1}∥|D|^{−1}(V ϕ) ∗ ǔ∥_{q} ≤ C∥V ϕ∥_{1}∥|D|^{−1}(V ϕ)∥_{\frac{m}{m−1} ,w}∥u∥_{p}. (5.42)

For I_{1}(x), Hölder’s inequality implies

|I_{1}(x)| ≤ C (∫_{|y|≤1} |\frac{(V ϕ)(x − y)}{|y|^{m−2}} |^{q′} dy)^{1/q′} (∫_{|y|≤1} |MH(M_{∗})(|y|)|^{q}dy)^{1/q} .

The second factor on the right is bounded by C∥|D|^{−1}(V ϕ)∥_{\frac{m}{m−1} ,w}∥u∥_{p} as in (5.42) and q′ < \frac{m}{m−1} < \frac{m}{2} < p. It follows by Minkowski’s inequality that

∥I_{1}∥_{p} ≤ C∥V ϕ∥_{p}∥u∥_{p} (∫_{|y|≤1} \frac{dy}{|y|^{(m−2)q′}} )^{1/q′} ≤ C∥V ϕ∥_{p}∥u∥_{p}.

Thus, we have ∥Z_{∗}^{00}u∥_{p} ≤ C∥u∥_{p} for \frac{m}{2} < p < m. The boundary term of (5.41) is, by virtue of (3.7) and that c_{0} = (m − 2)^{−1}, equal to

\frac{−i}{2π} ∫_{ℝ} M_{∗}(r)dr = \frac{−i}{πω_{m−1}} ∫_{ℝ^{m}} (∫_{ℝ^{m}} \frac{|D|^{−1}(V ϕ)(y)}{|x − y|^{m−1}}dy) u(x)dx

= \frac{−iΓ (\frac{m}{2})}{\sqrt{π}Γ (\frac{m−1}{2})} ∫_{ℝ^{m}} |D|^{−2}(V ϕ)(x)u(x)dx = \frac{iΓ (\frac{m}{2})}{\sqrt{π}Γ (\frac{m−1}{2})}⟨ϕ, u⟩.  (5.43)

Inserting this into the right of (5.8) for j = k = 0, we see the contribution of the boundary term to Z_{s1}(ϕ)u is given by

\frac{2c_{0}C_{0}Γ (\frac{m}{2})}{\sqrt{π}Γ (\frac{m−1}{2})} ∫_{ℝ^{m}} \frac{V ϕ(y)}{|x − y|^{m−2}} dy⟨ϕ, u⟩ = −\frac{Γ (\frac{m−2}{2})}{\sqrt{π}Γ (\frac{m−1}{2})}|ϕ⟩⟨ϕ, u⟩.

This proves the first statement. If Z_{s1} ∈ B(L^{p}) for some \frac{m}{2} < p < m, (5.39) implies ϕ ⊗ ϕ ∈ B(L^{p}) for this p. Then, (3.8) implies that ϕ must satisfy ⟨ϕ, V ⟩ = 0 and ϕ ⊗ ϕ ∈ B(L^{p}) for all \frac{m}{m−1} < p < m. Then, Z_{s1} ∈ B(L^{p}) must be satisfied for all \frac{m}{2} < p < m and, hence, for all 1 < p < m by Lemma 5.4 and interpolation.

We finally study Z_{1s}(ϕ) in L^{p}(ℝ^{m}) for p > m. If Z_{1s}(ϕ) ∈ B(L^{p}(ℝ^{m})) for some p > m, then Lemma 5.5 implies ϕ ∈ E_{0}. Thus, assume ϕ ∈ E_{0} in the following lemma. The following lemma proves statements (3) of Theorem 1.4 and Theorem 1.5 for odd m ≥ 7.

Lemma 5.6. Let m ≥ 5 be odd, p > m and ϕ ∈ E_{0}. Then:

(1) For a constant C_{p} > 0, ∥Z_{s1}(ϕ)u + |ϕ⟩⟨ϕ|)u∥_{p} ≤ C∥u∥_{p}.

(2) If Z_{1s}(ϕ) is bounded in L^{p}(ℝ^{m}) for some p > m, then ϕ ∈ E_{1}. In this case Z_{1s} is bounded in L^{p}(ℝ^{m}) for all 1 < p < ∞.

Proof. Considering that ∫_{ℝ} r^{j+1}e^{−iλr}M_{∗}(r)dr = i^{j+1} (∫_{ℝ} e^{−iλr}M_{∗}(r)dr)^{(j+1)}, we apply integration by parts to (5.22). Then, for k ≥ 1, we have

K^{(j,k)}(ρ) = \frac{(−i)^{j+1}}{2π} ∫_{0}^{∞} (e^{iλρ}λ^{j+k}F (λ))^{(j+1)} (∫_{ℝ} e^{−iλr}M_{∗}(r)dr) dλ (5.44)

and, if k = 0, additional boundary term which is given by virtue of (5.43) by

\frac{(−i)^{j+1}j!}{2π} ∫_{ℝ} M_{∗}(r)dr = \frac{i(−i)^{j}j!Γ (\frac{m}{2})}{\sqrt{π}Γ (\frac{m−1}{2})}⟨ϕ, u⟩, j = 0, …, \frac{m − 3}{2}. (5.45)

Denote the right of (5.44) by K̃^{(j,0)}(ρ) when k = 0. Then,

\frac{K^{(j,k)}(ρ)}{ρ^{m−2−k}}≤_{| · |} C (1 + \frac{1}{ρ^{m−2}} ) MH(M_{∗})(ρ), 0 ≤ j, k ≤ \frac{m − 3}{2}  (5.46)

and the same for K̃^{(j,0)}(ρ). We split Z_{s1}^{jk}u as previously:

Z_{s1}^{jk}u(x) = (∫_{|x−y|≤1} + ∫_{|x−y|>1}) \frac{V ϕ(y)K^{(j,k)}(|x − y|)}{|x − y|^{m−2−k}}dy = I_{1}(x) + I_{2}(x).

We estimate I_{2}(x) by using (5.46) for ρ ≥ 1, that ρ^{m−1} is A_{p} weight for p > m, (3.8) for ϕ ∈ E_{0} and the Calderón-Zygmund theory. This yields

∥I_{2}∥_{p} ≤ ∥V ϕ∥_{1} (∫_{1}^{∞} |MH(M_{∗})(ρ)|^{p}ρ^{m−1}dρ)^{1/p}
≤ ∥V ϕ∥_{1}∥|D|^{−1}(V ϕ) ∗ u∥_{p} ≤ C∥V ϕ∥_{1}∥u∥_{p}.  (5.47)

Hölder’s inequality and (5.46) for ρ ≤ 1 imply

|I_{1}(x)| ≤ C (∫_{|y|≤1} |\frac{(V ϕ)(x − y)}{|y|^{m−2}} |^{p′} dy)^{1/p′} (∫_{|y|≤1} |MH(M_{∗})(|y|)|^{p}dy)^{1/p} .

The second factor on the right is bounded by C∥u∥_{p} as in (5.47). Since p′ < \frac{m}{m−1} < m < p, it follows by Minkowski’s inequality that

∥I_{1}∥_{p} ≤ C∥V ϕ∥_{p}∥u∥_{p} (∫_{|y|≤1} \frac{dy}{|y|^{(m−2)p′}} )^{1/p′} ≤ C∥V ϕ∥_{p}∥u∥_{p}.

Thus, Z_{s1}^{jk} ∈ B(L^{p}(ℝ^{m})) for p > m if k ≥ 1 and the same for the operator Z̃_{s1}^{j0} produced by K̃^{(j,0)}(ρ). The contribution of boundary terms (5.45) to Z_{s1}(ϕ) is given by using the constants C_{j} of (2.2) by

2i ∑_{j=0}^{\frac{m−3}{2}} C_{0}C_{j}(−1)^{j+1}ω_{m−1} (∫_{ℝ^{d}} \frac{(V ϕ)(y)}{|x − y|^{m−2}} dy) \frac{i(−i)^{j}j!Γ (\frac{m}{2})}{\sqrt{π}Γ (\frac{m−1}{2})}⟨ϕ, u⟩

= −D̃_{m}|ϕ⟩⟨ϕ, u⟩, D̃_{m} = ∑_{j=0}^{\frac{m−3}{2}} \frac{(m − 3 − j)!}{2^{m−3−j} (\frac{m−3}{2})! (\frac{m−3}{2} − j)!}. (5.48)

The constant D̃_{m} can be elementarily computed and with n = \frac{m−3}{2}

D̃_{m} = ∑_{k=0}^{n} \frac{1}{2^{2n−k}} (\begin{matrix}2n − k \\ n − k \end{matrix}) = ∑_{k=0}^{n} \frac{1}{2^{n+k}} (\begin{matrix}n + k \\ k \end{matrix}) = 1.

(see also page 167 of [12].) This proves statement (1). We omit the proof of (2) which is similar to the corresponding statement of Lemma 5.5.

Since Z_{s1}u = ∑_{i=1}^{n} Z_{s1}(ϕ_{j}) for the orthonormal basis of E, the combination of lemmas in this section proves Theorems 1.4 and 1.5 for odd m.

6 Proof of Theorem 1.5 for even m ≥ 6

For proving Theorem 1.5 for even dimensions m ≥ 6 we need study Z_{s} and Z_{log} of (3.26) and (3.27). Since Z_{log} may be studied in a way similar to but simpler than that for Z_{s}, we shall be mostly concentrated on Z_{s} and only briefly comment on Z_{log} at the end of the section. As in odd dimensions we take the real orthonormal basis {ϕ_{1}, …, ϕ_{d}} of E and define, for ϕ ∈ E,

Z_{s}(ϕ)u = \frac{i}{π} ∫_{0}^{∞} G_{0}(λ)|V ϕ⟩⟨ϕV |(G_{0}(λ) − G_{0}(−λ))F (λ)λ^{−1}dλ.  (6.1)

Then, we have

Z_{s}u = ∑_{j=1}^{d} Z_{s}(ϕ_{j})u

and we study Z_{s}(ϕ) for ϕ ∈ E. In this section we choose and fix a ϕ ∈ E arbitrarily and write M(r) = M(r, V ϕ ∗ ǔ).

We wish to apply the argument for odd dimensions also to even dimensions as much as possible and, we express Z_{s}(ϕ) as a superposition of operators which are of the same form as those studied in odd dimensions except scaling. We set ν = (m − 2)/2. Define for a > 0

M^{a}(r) = M(r/(1 + 2a))  (6.2)

and, for j, k = 0, …, ν and a, b > 0,

Q_{jk}^{a,b}(ρ) = \frac{(−1)^{j+1}}{2π(1 + 2a)^{j+2}} ∫_{0}^{∞} λ^{j+k−1}e^{iλ(1+2b)ρ}F(r^{j+1}M^{a})(λ)F (λ)dλ. (6.3)

As in (5.22), we may express Q_{jk}^{a,b}(ρ) by using M_{∗}(r) and increase the factor λ^{j+k−1} of (6.3) to λ^{j+k}:

Q_{jk}^{a,b}(ρ) = \frac{(−1)^{j+1}}{2π(1 + 2a)^{j+2}} ∫_{0}^{∞} λ^{j+k}e^{iλ(1+2b)ρ}F(r^{j+1}M_{∗}^{a})(λ)F (λ)dλ. (6.4)

When j = 0, we also use M̃(r) of (2.15) to express Q_{0k}^{a,b}(ρ) as follows:

Q_{0k}^{a,b}(ρ) = \frac{i}{2π(1 + 2a)^{2}} ∫_{0}^{∞} λ^{k}e^{iλ(1+2b)ρ}F(M̃^{a})(λ)F (λ)dλ.  (6.5)

Lemma 6.1. Let Q_{jk}^{a,b}(ρ) be defined by (6.3), (6.4) or (6.5). Then,

Z_{s}(ϕ)u(x) = \frac{2i}{ω_{m−1}} ∑_{j,k=0}^{ν} T_{j}^{(a)}T_{k}^{(b)} [∫_{ℝ^{m}} \frac{(V ϕ)(x − y)Q_{jk}^{a,b}(|y|)}{|y|^{m−2−k}}dy] . (6.6)

Proof. We apply (2.18) for ⟨V ϕ, (G_{0}(λ) − G_{0}(−λ))u⟩ and (2.6) for G_{0}(λ) in (6.1). We see that Z_{s}(ϕ)u(x) is the integral with respect to λ ∈ (0, ∞) of

\frac{i}{π} ∑_{j,k=0}^{ν} T_{j}^{(a)}T_{k}^{(b)} [\frac{(−1)^{j+1}λ^{j+k−1}}{(1 + 2a)^{j+2}ω_{m−1}} (\frac{e^{iλ(1+2b)|y|}}{|y|^{m−2−k}} ∗ V ϕ) F(r^{j+1}M^{a})(λ)] F (λ).

Integrating with respect to λ first yields (6.6).

We define, for 0 ≤ j, k ≤ ν and a, b > 0, that

Z^{jk}(ϕ)u(x) = \frac{2i}{ω_{m−1}} T_{j}^{(a)}T_{k}^{(b)} [Z_{a,b}^{jk}(ϕ)u(x)] ,  (6.7)

Z_{a,b}^{jk}(ϕ)u(x) = ∫_{ℝ^{m}} \frac{(V ϕ)(x − y)Q_{jk}^{a,b}(|y|)}{|y|^{m−2−k}}dy.  (6.8)

Lemma 6.1 implies Z_{s}(ϕ)u = ∑ Z^{jk}(ϕ)u. In what follows we often write Z^{jk}u and Z_{a,b}^{jk} respectively for Z^{jk}(ϕ)u and Z_{a,b}^{jk}(ϕ).

6.1 Estimate of ∥Z^{jk}u∥_{p} for (j, k) ≠ (ν, ν).

We estimate Z^{jk} for the case (j, k) ≠ (ν, ν) first, postponing the case (j, k) = (ν, ν) to the next subsection. As we shall see, the argument used for odd dimensions applies to Z^{jk} if (j, k) ≠ (ν, ν) modulo superpositions and scalings.

Lemma 6.2. With suitable constants C > 0, followings are majorants of Q_{jk}^{a,b}(ρ) for 0 ≤ k, j ≤ ν which satisfy the attached conditions respectively:

(1) C \frac{\{MH(r^{j+1}M^{a})\}((1 + 2b)ρ)}{(1 + 2a)^{j+2}}, if j + k ≥ 1.  (6.9)

(2) C \frac{MH(M̃^{a})((1 + 2b)ρ)}{(1 + 2a)^{2}}, if j = 0.  (6.10)

(3) C ∑_{l=0}^{k+1} \frac{MH(r^{j+l+1}M^{a})((1 + 2b)ρ)}{(1 + 2a)^{j+2}(1 + 2b)^{k+1}ρ^{k+1}}, if 2 ≤ j ≤ ν.  (6.11)

(4) C \frac{MH(r^{2}M^{a})((1 + 2b)ρ)}{(1 + 2a)^{j+2}}\{(1 + 2b)^{j−1}ρ^{j−1} + 1\}, if 1 ≤ j. (6.12)
(5) C \frac{MH(rM^{a})((1 + 2b)ρ)}{(1 + 2a)^{j+2}}\{(2b + 1)^{j}ρ^{j} + 1\}, for all j, k.  (6.13)

Proof. Define Φ_{jk}(λ) = λ^{j+k−1}F (λ). If j + k ≥ 1, Φ_{jk} ∈ C_{0}^{∞}(ℝ) and Lemma 2.9 implies Q_{jk}^{a,b}(ρ) = (−1)^{j+1}(1 + 2a)^{−(j+2)}{(FΦ_{jk}) ∗ H(r^{j+1}M^{a})}((1 + 2b)ρ). Then, (6.9) follows by applying (2.24). Likewise we have (6.10) from (6.5). If j ≥ 2, we apply integration by parts k +1 times to (6.3) using that e^{iλ(1+2b)ρ} = (i(1 + 2b)ρ)^{−(k+1)}∂_{λ}^{k+1}e^{iλ(1+2b)ρ} then, without boundary terms,

Q_{jk}^{a,b}(ρ) = ∑_{l=0}^{k+1} \frac{(−1)^{j+1}}{2π(1 + 2a)^{j+2}} (\frac{1}{−i(1 + 2b)ρ})^{k+1} (\begin{matrix}k + 1 \\ l \end{matrix})

× ∫_{0}^{∞} e^{iλ(1+2b)ρ}Φ_{jk}(λ)^{(k+1−l)}F((−i)^{l}r^{j+l+1}M^{a})(λ)dλ  (6.14)

and (6.11) follows as previously. If j ≥ 1, we may apply integration by parts to (6.3) by using that F(r^{j+1}M^{a})(λ) = i^{j−1}{F(r^{2}M^{a})(λ)}^{(j−1)}. Then

Q_{jk}^{a,b}(ρ) = i^{j−1} ∫_{0}^{∞} \frac{(λ^{j+k−1}F (λ)e^{iλ(1+2b)ρ})^{(j−1)} F(r^{2}M^{a})(λ)}{2π(1 + 2a)^{j+2}}dλ (6.15)

and (6.12) follows. Apply another integration by parts in (6.15). No boundary term appears as F(rM^{a})(0) = 0, and we obtain (6.13).

6.1.1 Estimate for 1 < p < \frac{m}{m−1}

Define for 0 ≤ σ ≤ m − 1 and 1 < p < \frac{m}{m−1}:

N_{σ}^{a,b}(u) = (∫_{0}^{∞} |MH(r^{σ}M^{a})((1 + 2b)ρ)|^{p}ρ^{m−1−p(m−1)}dρ)^{1/p} . (6.16)

Lemma 6.3. For any \frac{m}{1+σ} ≤ q ≤ ∞, we have

N_{σ}^{a,b} ≤ C \frac{(1 + 2b)^{m−1− \frac{m}{p}}}{(1 + 2a)^{m−1− \frac{m}{p} −σ}} (∥V ϕ∥_{1} + ∥V ϕ∥_{q})∥u∥_{p}.  (6.17)

Proof. Change variable ρ by (1 + 2b)^{−1}ρ first. Since ρ^{m−1−p(m−1)} is an A_{p}- weight,

N_{σ}^{a,b} = (1 + 2b)^{m−1− \frac{m}{p}} (∫_{0}^{∞} |MH(r^{σ}M^{a})(ρ)|^{p}ρ^{m−1−p(m−1)}dρ)^{1/p}

≤ C \frac{(1 + 2b)^{m−1− \frac{m}{p}}}{(1 + 2a)^{m−1− \frac{m}{p} −σ}} (∫_{0}^{∞} |M(r)|^{p}r^{m−1−p(m−1−σ)}dr)^{1/p} . (6.18)

Denote by I the integral on (6.18). Let κ = m − 1 − σ. If κ = 0, then I ≤ C∥V ϕ ∗ u∥_{p} ≤ C∥V ϕ∥_{1}∥u∥_{p} and (6.17) follows. Let 0 < κ ≤ m − 1. Split I into integral over 0 < r < 1 and r > 1 and use r^{m−1−pκ} ≤ r^{m−1} for r ≥ 1. Then, we have I ≤ C(∥|x|^{−κ}(V ϕ ∗ u)(x)∥_{L^{p}(|x|<1)} + ∥V ϕ∥_{1}∥u∥_{p}). Take κ′ such that κ < κ′ < m and apply Hölder’s and Young’s inequalities for the integral over |x| ≤ 1. We obtain with q = \frac{m}{m−κ′} ∈ [\frac{m}{1+σ} , ∞] that

I ≤ C(∥|x|^{−κ}∥_{L^{\frac{m}{κ′}} (|x|≤1)}∥V ϕ∥_{q} + ∥V ϕ∥_{1})∥u∥_{p}.  (6.19)

This completes the proof.

Lemma 6.4. Suppose 1 < p < \frac{m}{m−1}. Then, for 2 ≤ j ≤ ν and 0 ≤ k ≤ ν such that (j, k) ≠ (ν, ν),

∥Z^{jk}u∥_{p} ≤ C∥u∥_{p}, u ∈ C_{0}^{∞}(ℝ^{m}).  (6.20)

Proof. Minkowski’s and Young’s inequality imply

∥Z^{jk}u∥_{p} ≤ 2ω_{m−1}^{−1}∥V ϕ∥_{1} · T_{j}^{(a)}T_{k}^{(b)} [∥|x|^{2+k−m}Q_{jk}^{a,b}∥_{p}] .  (6.21)

We apply (6.11) to estimate Q_{jk}^{a,b}(|x|). Then, since σ ≡ j + l + 1 ≤ m − 1 for (j, k) ≠ (ν, ν), Lemma 6.3 implies

∥|x|^{2+k−m}Q_{jk}^{a,b}∥_{p} ≤ C(1 + 2a)^{\frac{m}{p} −(m−k−1)}(1 + 2b)^{m−2− \frac{m}{p} −k}∥u∥_{p}. (6.22)

We plug this to (6.21) and use m − k − 1 ≥ j + 2. Then,

∥Z^{jk}u∥_{p} ≤ C_{mjk}T_{j}^{(a)}T_{k}^{(b)}[(1 + 2a)^{\frac{m}{p} −(j+2)}(1 + 2b)^{m−2− \frac{m}{p} −k}]∥u∥_{p}

≤ C∥u∥_{p} (∫_{0}^{∞} \frac{(1 + 2a)^{\frac{m}{p} −(j+2)}}{(1 + a)^{(2ν−j+ \frac{1}{2} )}} \frac{da}{\sqrt{a}}) (∫_{0}^{∞} \frac{(1 + 2b)^{m−2− \frac{m}{p} −k}}{(1 + b)^{(2ν−k+ \frac{1}{2} )}} \frac{db}{\sqrt{b}}) .

Counting powers show that the integrals are finite and the lemma follows.

Documenta Mathematica 21 (2016) 391–443

Remarks on L^{p}-Boundedness of Wave Operators for … 427 As in odd dimensions we use the cancellation in

Z^{0k}u + Z^{1k}u

= \frac{2i}{ω_{m−1}} ∫_{ℝ^{m}} \frac{(V ϕ)(x − y)}{|y|^{m−2−k}}T_{k}^{(b)}(T_{0}^{(a)}Q_{0k}^{a,b}(|y|) + T_{1}^{(a)}Q_{1k}^{a,b}(|y|))dy (6.23)

and obtain the following lemma. Lemma 6.5. For 1 < p < \frac{m}{m−1}, there exists a constant C > 0 such that

∥(Z^{(0,k)} + Z^{(1,k)})u∥_{p} ≤ C∥u∥_{p}, k = 0, …, ν.  (6.24)

Proof. We apply integration by parts k + 1 times to (6.5) and (6.3) as in the proof of (6.11). This produces

Q_{0k}^{a,b}(ρ) = \frac{−i^{k}k!(FM̃^{a})(0)ω_{m−1}}{2π(1 + 2a)^{2}(1 + 2b)^{k+1}ρ^{k+1}} − \frac{i^{k}ω_{m−1}}{2π} ∑_{l=0}^{k+1} C_{k+1,l}Q_{0k,l}^{a,b}(ρ), (6.25)
Q_{1k}^{a,b}(ρ) = \frac{i^{k+1}k!F(r^{2}M^{a})(0)ω_{m−1}}{2π(1 + 2a)^{3}(1 + 2b)^{k+1}ρ^{k+1}} + \frac{i^{k+1}ω_{m−1}}{2π} ∑_{l=0}^{k+1} C_{k+1,l}Q_{1k,l}^{a,b}(ρ),(6.26)

where Q_{0k,l}^{a,b}(ρ) and Q_{1k,l}^{a,b}(ρ) are given and estimated as follows:

Q_{0k,l}^{a,b}(ρ) = ∫_{0}^{∞} \frac{e^{iλ(1+2b)ρ}(λ^{k}F (λ))^{(k+1−l)}(F((−ir)^{l}M̃^{a})(λ))}{(1 + 2a)^{2}(1 + 2b)^{k+1}ρ^{k+1}}dλ
≤_{| · |} C \frac{MH(r^{l}M̃^{a})((1 + 2b)ρ))}{(1 + 2a)^{2}(1 + 2b)^{k+1}ρ^{k+1}} ,  (6.27)
Q_{1k,l}^{a,b}(ρ) = (−i)^{l} ∫_{0}^{∞} \frac{e^{iλ(1+2b)ρ}(λ^{k}F (λ))^{(k+1−l)}F(r^{2+l}M^{a})(λ))}{(1 + 2a)^{3}(1 + 2b)^{k+1}ρ^{k+1}}dλ
≤_{| · |} C \frac{MH(r^{2+l}M^{a})((1 + 2b)ρ)}{(1 + 2a)^{3}(1 + 2b)^{k+1}ρ^{k+1}}.  (6.28)

Eqn.(2.16) shows F(M̃^{a})(0) = F(r^{2}M^{a})(0) = (1 + 2a)^{3} ∫_{0}^{∞} r^{2}M(r)dr and

T_{1}^{(a)}[i] = T_{0}^{(a)}[(1 + 2a)] = (m − 3)^{−1}

It follows that the sum of the superposition via T_{0}^{(a)} of the boundary term of (6.25) and that via T_{1}^{(a)} of (6.26) vanishes:

\frac{i^{k}k!}{(1 + 2b)^{k+1}ρ^{k+1}} (∫_{0}^{∞} r^{2}M(r)dr) (T_{1}^{(a)}[i] − T_{0}^{(a)}[(1 + 2a)]) = 0. (6.29)

For 1 < p < \frac{m}{m−1}, ρ^{m−1−p(m−1)} is an A_{p} weight on ℝ and we have the identity:

M̃^{a}(r) = ∫_{r}^{∞} sM^{a}(s)ds = (1 + 2a)^{2}M̃((1 + 2a)^{−1}r).  (6.30)

Documenta Mathematica 21 (2016) 391–443

Then, Lemma 2.7, (6.30), change of variable and Hardy’s inequality imply

∥\frac{Q_{0k,l}^{a,b}(|x|)}{|x|^{m−k−2}} ∥_{p} ≤ \frac{C(1 + 2b)^{m−1− \frac{m}{p}}}{(1 + 2a)^{2}(1 + 2b)^{k+1}} (∫_{0}^{∞} |r^{l}M̃^{a}(r)|^{p}r^{m−1−p(m−1)}dr)^{1/p}

≤ \frac{C(1 + 2a)^{\frac{m}{p} −(m−1−l)}}{(1 + 2b)^{\frac{m}{p} −(m−k−2)}} (∫_{0}^{∞} |M(r)|^{p}r^{m−1−p(m−3−l)}dr)^{1/p} .  (6.31)

The integral is similar to the integral which appeared in (6.18) and we remark m − 3 − l ≥ 0 for m ≥ 6. Thus, applying (6.19) with σ = l + 2, we obtain

(6.31) ≤ \frac{C(1 + 2a)^{\frac{m}{p} −(m−k−2)}}{(1 + 2b)^{\frac{m}{p} −(m−k−2)}} (∥V ϕ∥_{1} + ∥V ϕ∥\frac{m}{3} )∥u∥_{p}, 0 ≤ l ≤ k + 1. (6.32)

Counting the powers of a and b, we thus have from (6.32) that

T_{0}^{(a)}T_{k}^{(b)} [∥|x|^{2+k−m}Q_{0k,l}^{a,b}∥_{p}] ≤ C∥u∥_{p}. 0 ≤ l ≤ k + 1.  (6.33)

Entirely similarly, starting from (6.28), we obtain

∥\frac{Q_{1k,l}^{a,b}(|x|)}{|x|^{m−k−2}} ∥_{p} ≤ \frac{C(1 + 2b)^{m−1− \frac{m}{p}}}{(1 + 2a)^{3}(1 + 2b)^{k+1}} (∫_{0}^{∞} |r^{2+l}M^{a}(r)|^{p}r^{m−1−p(m−1)}dr)^{1/p}

≤ \frac{C(1 + 2a)^{\frac{m}{p} −(m−k−1)}}{(1 + 2b)^{\frac{m}{p} −(m−k−2)}} (∥V ϕ∥_{1} + ∥V ϕ∥\frac{m}{3} )∥u∥_{p}, 0 ≤ l ≤ k + 1.  (6.34)

The extra decaying factor (1 + 2a)^{−1} of (6.34) compared to (6.32) cancels the extra increasing factor (1 + a) of T_{1}^{(a)} compared to T_{0}^{(a)} and we have

T_{1}^{(a)}T_{k}^{(b)} [∥|x|^{k+2−m}Q_{0k,l}^{a,b}(|x|)∥_{p}] ≤ C∥u∥_{p}, 0 ≤ l ≤ k + 1.  (6.35)

In view of (6.23), (6.25), (6.26) and (6.29), (6.33) and (6.35) with the help of Young’s and Minkowski’s inequalities imply the lemma.

6.1.2 Estimate for \frac{m}{3} < p < \frac{m}{2}

The following lemma together with Lemma 6.4 and Lemma 6.5 will prove that ∑_{(j,k)≠(ν,ν)} Z^{jk} is bounded in L^{p}(ℝ^{m}) for 1 < p < \frac{m}{2}.

Lemma 6.6. Let \frac{m}{3} < p < \frac{m}{2}. Then, for (j, k) ≠ (ν, ν),

∥Z^{jk}u∥_{p} ≤ C_{p}∥u∥_{p}  (6.36)

for a constant C_{p} > 0 independent of u ∈ C_{0}^{∞}(ℝ^{m}).

Proof. Except the superposition the proof is virtually the repetition of that of statement (2) of Lemma 5.3.

(1) Let j ≥ 1 first. Since ρ^{m−1−2p} is A_{p} weight for \frac{m}{3} < p < \frac{m}{2}, we have

(∫_{0}^{∞} |\{MH(r^{2}M^{a})\}(ρ)|^{p}ρ^{m−1−2p}dρ)^{\frac{1}{p}} ≤ C(1 + 2a)^{\frac{m}{p}} ∥V ϕ∥_{1}∥u∥_{p}. (6.37)

Splitting the integral of (6.8) we define

Z_{jk}^{a,b}u(x) = (∫_{|y|< \frac{1}{1+2b}} + ∫_{|y|> \frac{1}{1+2b}} ) \frac{(V ϕ)(x − y)Q_{jk}^{a,b}(|y|)}{|y|^{m−2−k}}dy = I_{1}(x) + I_{2}(x).

(6.38)

For I_{1}(x), we estimate |y|^{−(m−k−2)} ≤ |y|^{−(m−2)} for |y| ≤ 1 and apply Hölder’s inequality. Then

∥I_{1}∥_{p} ≤ ∥∫_{|y|≤ \frac{1}{2b+1}} \frac{|(V ϕ)(x − y)|^{p′}dy}{|y|^{p′(m−4)}} ∥_{p/p′}^{1/p′} (∫_{|y|≤ \frac{1}{2b+1}} |\frac{Q_{jk}^{a,b}(|y|)}{|y|^{2}} |^{p} dy)^{1/p}

Minkowski’s inequality implies that the first factor on the right is bounded by C∥V ϕ∥_{p}(1 + 2b)^{m−4− \frac{m}{p′}} and \frac{m}{p′} − (m − 4) > 1. For the second factor, we apply (6.12) for (1 + 2b)ρ < 1 and then (6.37). We obtain

∥I_{1}∥_{p} ≤ C(1 + 2a)^{\frac{m}{p} −j−2}(1 + 2b)^{1− \frac{m}{p}} ∥V ϕ∥_{1}∥V ϕ∥_{p}∥u∥_{p}.  (6.39)

By Young’s inequality ∥I_{2}∥_{p} ≤ C∥V ϕ∥_{1}∥|x|^{2+k−m}Q_{jk}^{a,b}(|x|)∥_{L^{p}((1+2b)|x|>1)}. For the second factor, we use (6.12) for (1 + 2b)ρ ≥ 1 and, after changing the variables ρ → (1 + 2b)^{−1}ρ, we estimate ρ^{−(m−2−k−(j−1))} ≤ ρ^{−2} for ρ ≥ 1 (here we used (j, k) ≠ (ν, ν)) and apply (6.37) once more. Then,

∥I_{2}∥_{p} ≤ C∥V ϕ∥_{1} \frac{(1 + 2b)^{m−2−k− \frac{m}{p}}}{(1 + 2a)^{j+2}} (∫_{1}^{∞} |\{MH(r^{2}M^{a})\}(ρ)|^{p}ρ^{m−1−2p}dρ)^{\frac{1}{p}}

≤ C(1 + 2a)^{\frac{m}{p} −j−2}(1 + 2b)^{m−2−k− \frac{m}{p}} ∥V ϕ∥_{1}^{2}∥u∥_{p}.  (6.40)

Since m − 2 − k ≥ 1 and (1 + 2a)^{\frac{m}{p} −j−2}(1 + 2b)^{m−2−k− \frac{m}{p}} is summable by T_{j}^{(a)}T_{k}^{(b)}, (6.39) and (6.40) imply

∥Z^{jk}u∥_{p} ≤ C∥V ϕ∥_{1}(∥V ϕ∥_{1} + ∥V ϕ∥_{p})∥u∥_{p}.  (6.41)

(2) When j = 0, we apply the argument in the proof in (1) for estimating Q_{0k}^{a,b} but by using (6.10) in stead of (6.12). Then, by the help of (6.30) and Hardy’s inequality, it leads to estimates (6.39) and (6.40) and, hence, to the desired (6.36) for Z^{0k}. This completes the proof of the lemma.

6.1.3 Estimate for m/2 < p < m and for p > m.

We now estimate Z^{jk}, (j, k) ≠ (ν, ν), in L^{p}(ℝ^{m}) for \frac{m}{2} < p < m and for p > m. As in odd dimensions, Z^{00} will not in general be bounded in L^{p}(ℝ^{m}) when \frac{m}{2} < p < m and likewise for all Z^{0k}, k = 0, …, \frac{m−2}{2} when p > m. Elementary computations using

z^{−k} = \frac{1}{Γ(k)} ∫_{0}^{∞} e^{−zt}t^{k−1}dt, ℜz > 0, k > 0

and the formula (2.5) for C_{m,j}ω_{m−1} we obtain the following lemma.

Lemma 6.7.

(1) We have T_{j}^{(a)}[1] = (m − 3 − j)!/(m − 2)!.

(2) For k ≥ 1 and j = 0, ⋯, ν, T_{j}^{(a)}[(1 + 2a)^{−k}] is given by

\frac{(−i)^{j}2^{m−1}Γ(2ν − j + k)}{(m − 2)!Γ(k)} (\begin{matrix}ν \\ j \end{matrix}) . ∫_{1}^{∞} \frac{(x^{2} − 1)^{k−1}}{(x^{2} + 1)^{2ν−j+k}} dx  (6.42)

Lemma 6.8. Let \frac{m}{2} < p < m and ϕ ∈ E. Then:

(1) If (j, k) ≠ (0, 0) or (j, k) ≠ (ν, ν), Z^{jk} is bounded in L^{p}(ℝ^{m}):

∥Z^{jk}u∥_{p} ≤ C∥u∥_{p}, u ∈ C_{0}^{∞}(ℝ^{m})  (6.43)

(2) There exists a constant C > 0 such that for u ∈ C_{0}^{∞}(ℝ^{m}), we have

∥Z^{00}u + D_{m}|ϕ⟩⟨ϕ, u⟩∥_{p} ≤ C∥u∥_{p},  (6.44)
D_{m} = \frac{2^{m}Γ (\frac{m}{2})}{\sqrt{π}Γ (\frac{m−1}{2})} ∫_{1}^{∞}(1 + x^{2})^{m−1}dx.  (6.45)

If Z^{00}(ϕ) is bounded in L^{p}(ℝ^{m}) for some \frac{m}{2} < p < m then ϕ ∈ E_{0}. In this case Z^{00}(ϕ) is bounded in L^{p}(ℝ^{m}) for all \frac{m}{2} < p < m.

Proof. (1) Split Z_{a,b}^{jk}u(x) as in (6.38) and apply the argument thereafter to I_{1}(x) and I_{2}(x) by using the estimate (6.13). Since m − 2 − (k + j) ≥ 1 and ρ^{m−1−p} is an A_{p} weight for \frac{m}{2} < p < m, we have, as in (6.40),

∥I_{2}∥_{p} ≤ C \frac{(1 + 2b)^{m−2−k− \frac{m}{p}}}{(1 + 2a)^{j+2− \frac{m}{p}}} ∥V ϕ∥_{1}^{2}∥u∥_{p}.  (6.46)

For dealing with I_{1}(x), we estimate |y|^{−(m−2−k)} ≤ |y|^{m−2} for |y| ≤ 1 as previously but now decompose |y|^{−(m−2)} = |y|^{−(m−3)} · |y|^{−1}, remarking that (m − 3)p′ < m and p/p′ > 1. Then, we obtain as in (6.39) that

∥I_{1}∥_{p} ≤ \frac{(1 + 2a)^{\frac{m}{p} −j−2}}{(1 + 2b)^{\frac{m}{p}}} ∥V ϕ∥_{1}∥V ϕ∥_{p}∥u∥_{p}.  (6.47)

Documenta Mathematica 21 (2016) 391–443

Summing up (6.46) and (6.47) by T_{j}^{(a)}T_{k}^{(b)}, we obtain (6.43).

(2) Let j = k = 0. We apply integration by parts to (6.4).

Q_{00}^{a,b}(ρ) = \frac{−i}{2π(1 + 2a)^{2}} ∫_{0}^{∞} e^{iλ(1+2b)ρ}F(M_{∗}^{a})′(λ)F (λ)dλ

= \frac{i}{2π} ∫_{ℝ} \frac{M_{∗}^{a}(r)}{(1 + 2a)^{2}} dr + \frac{i}{(1 + 2a)^{2}} ∫_{0}^{∞}(F (λ)e^{iλ(1+2b)ρ})′F(M_{∗}^{a})(λ)dλ. (6.48)

Denote the second term on (6.48) by Q̃_{00}^{ab}(ρ) and by Z̃^{00} the operator produced by inserting Q̃_{00}^{ab}(ρ) for Q_{00}^{ab}(ρ) in (6.7). We have

Q̃_{00}^{a,b}(ρ)≤_{| · |} C \frac{MH(M_{∗}^{a})((1 + 2b)ρ)}{(1 + 2a)^{2}}(1 + (1 + 2b)ρ).  (6.49)

Let \frac{m}{2} < p < m. We split as in (6.38) and estimate I_{2} first:

Z̃^{00}u(x) = (∫_{|y|< \frac{1}{1+2b}} + ∫_{|y|≥ \frac{1}{1+2b}} ) \frac{(V ϕ)(x − y)Q̃_{00}^{a,b}(|y|)}{|y|^{m−2}}dy = I_{1}(x) + I_{2}(x).

We obtain

∥I_{2}∥_{p} ≤ C∥V ϕ∥_{1} \frac{(1 + 2b)^{m−2− \frac{m}{p}}}{(1 + 2a)^{2}} ∥\frac{MH(M_{∗}^{a})(|y|)}{|y|^{m−3}} ∥_{L^{p}(|y|>1)}

≤ C∥V ϕ∥_{1} \frac{(1 + 2b)^{m−2− \frac{m}{p}}}{(1 + 2a)^{2}} ∥\frac{1}{|y|^{m−3}} ∥_{L^{m}(|y|>1)} (∫_{0}^{∞} |M_{∗}^{a}(r)|^{q}r^{m−1}dr)^{\frac{1}{q}}

≤ C \frac{∥V ϕ∥_{1}(1 + 2b)^{m−2− \frac{m}{p}}}{(1 + 2a)^{2− \frac{m}{q}}} ∥|D|^{−1}(V ϕ)∥_{\frac{m}{m−1} ,∞}∥u∥_{p},  (6.50)

where we used Young’s inequality, (6.49) for (1 + 2b)ρ ≥ 1 and the change of variable (1 + 2b)ρ to ρ in the first step, Hölder’s inequality considering p^{−1} = m^{−1} + q^{−1} and that 1 is an A_{q} weight q = mp/(m − p) > m in the second and finally weak-Young’s inequality. For I_{1}, we apply Hölder’s and Minkowski’s inequalities and (6.50) and obtain

∥I_{1}∥_{p} ≤ C ∥(∫_{|y|≤ \frac{1}{1+2b}} |\frac{(V ϕ)(x − y)}{|y|^{m−2}} |^{q′} dy)^{\frac{1}{q′}} ∥_{p}

× (1 + 2b)^{− \frac{m}{p}} (1 + 2a)^{−2} (∫_{|y|≤1} |MH(M_{∗}^{a})(|y|)|^{q}dy)^{1/q}
≤ C(1 + 2b)^{− \frac{m}{p}} (1 + 2a)^{\frac{m}{p} −2}∥V ϕ∥_{p}∥|D|^{−1}(V ϕ)∥_{\frac{m}{m−1} ,∞}∥u∥_{p}. (6.51)

Summing (6.50) and (6.51) by T_{0}^{(a)}T_{0}^{(b)}, we obtain ∥Z̃^{(0,0)}u∥_{p} ≤ C∥u∥_{p}.

By virtue of (3.10) and (5.43), the contribution to Z^{00}u of the boundary term of (6.48) is given by

\frac{2i}{ω_{m−1}} T_{0}^{(a)}T_{0}^{(b)} [∫_{ℝ^{m}} \frac{(V ϕ)(y)dy}{|x − y|^{m−2}} · \frac{i}{2π} ∫_{ℝ} \frac{M_{∗}(r)}{(1 + 2a)}dr]

= −\frac{2}{C_{0}ω_{m−1}} T_{0}^{(a)}[(1 + 2a)^{−1}]T_{0}^{(b)}[1]\frac{Γ (\frac{m}{2})}{\sqrt{π}Γ (\frac{m−1}{2})}⟨ϕ, u⟩ϕ.  (6.52)

By using Lemma 6.7 and C_{0}ω_{m−1} = (m − 2)^{−1}. we can simplify (6.52) to −D_{m}⟨ϕ, u⟩ϕ with D_{m} given by (6.45) and (6.44) follows. The last statement follows as in the odd dimensional case, see the remark after Lemma 5.5.

Finally in this section we study Z^{jk}(ϕ)u for (j, k) ≠ (ν, ν) in L^{p}(ℝ^{m}) when p > m, assuming ϕ ∈ E_{0} by the same reason as in odd dimensions. We define

D_{m,j} = 2^{m} (\begin{matrix}ν \\ j \end{matrix}) \frac{Γ (\frac{m}{2})}{\sqrt{π}Γ (\frac{m−1}{2})} ∫_{1}^{∞} \frac{(x^{2} − 1)^{j}}{(x^{2} + 1)^{m−1}} dx, j = 0, …, ν. (6.53)

Lemma 6.9. Let m ≥ 6 be even and p > m. Suppose that ϕ ∈ E_{0}. Then:

(1) For (j, k) such that k ≠ 0 and (j, k) ≠ (ν, ν), Z^{jk} is bounded in L^{p}(ℝ^{m}).

(2) There exists a constant C > 0 such that

∥Z^{j0}u + D_{j,m}⟨ϕ, u⟩ϕ∥_{p} ≤ C∥u∥_{p}, j = 0, …, ν.  (6.54)

(3) If Z^{j0}(ϕ) is bounded in L^{p}(ℝ^{m}) for some 0 ≤ j ≤ ν and some m < p < ∞, then ϕ ∈ E_{1}. In this case, Z^{j0}(ϕ) is bounded in L^{p}(ℝ^{m}) for all 1 < p < ∞ and 0 ≤ j ≤ ν.

Proof. We apply integration by parts j + 1 times to (6.4):

Q_{jk}^{a,b}(ρ) = ∫_{0}^{∞} \frac{(−i)^{j+1}λ^{j+k}F (λ)e^{iλ(1+2b)ρ}∂_{λ}^{j+1}\{F(M_{∗}^{a})(λ)\}}{2π(1 + 2a)^{j+2}}dλ.  (6.55)

(1) If k ≥ 1, then no boundary terms appear and we have

Q_{jk}^{a,b}(ρ)≤_{| · |} \frac{CMH(M_{∗}^{a})((1 + 2b)ρ)}{(1 + 2a)^{j+2}}\{(1 + 2b)^{j+1}ρ^{j+1} + 1\}.  (6.56)

Observing that m−2−(k+j +1) ≥ 0 for (j, k) ≠ (ν, ν), that r^{m−1} is A_{p} weight on ℝ for p > m and that (m − 2 − k)p′ < m, we apply the argument used for proving (6.50) and (6.51) in the proof of the previous lemma and obtain

∥Z_{a,b}^{jk}u∥_{p} ≤ \frac{C∥V ϕ∥_{1}(1 + 2b)^{m−2−k− \frac{m}{p}}}{(1 + 2a)^{j+2− \frac{m}{p}}} ∥|D|^{−1}(V ϕ) ∗ u∥_{p}

+ \frac{C(1 + 2b)^{− \frac{m}{p}} ∥V ϕ∥_{p}}{(1 + 2a)^{j+2− \frac{m}{p}}} (∫_{|y|< \frac{1}{1+2b}} \frac{dy}{|y|^{(m−2−k)p′}} )^{\frac{1}{p′}} ∥|D|^{−1}(V ϕ) ∗ u∥_{p}. (6.57)

Documenta Mathematica 21 (2016) 391–443

Since ∫ (V ϕ)(x)dx = 0, ∥|D|^{−1}(V ϕ)∗u∥_{p} ≤ C∥u∥_{p} for any 1 < p < ∞ by virtue of (3.9) and the Calderón-Zygmund theory. It follows that

∥Z^{jk}u∥_{p} ≤ T_{j}^{(a)}T_{k}^{(b)}∥Z_{a,b}^{jk}u∥_{p} ≤ C∥u∥_{p}

for k ≥ 1 and (j, k) ≠ (ν, ν) and statement (1) is proved.

(2) If k = 0, then, j+1 times integration by parts in (6.55) produces the integral and boundary terms. The integral is bounded by (6.56) and the repetition of the argument in step (1) implies that its contribution to Z^{j0} is the operator which is bounded in L^{p}(ℝ^{m}) for all p > m. The boundary term may be expressed as follows by using (5.43) once more,

\frac{i^{j+1}j!}{2π(1 + 2a)^{j+1}} ∫_{ℝ} M_{∗}(r)dr = \frac{−i^{j+1}j!}{(1 + 2a)^{j+1}} \frac{Γ (\frac{m}{2})}{Γ (\frac{m−1}{2}) \sqrt{π}} ⟨ϕ, u⟩,  (6.58)

and its contribution Z^{j0}u may be computed as follows:

\frac{2i}{ω_{m−1}} T_{j}^{(a)}T_{0}^{(b)} [∫_{ℝ^{m}} \frac{(V ϕ)(y)dy}{|x − y|^{m−2}} (−\frac{i^{j+1}j!}{(1 + 2a)^{j+1}} )] \frac{Γ (\frac{m}{2})}{Γ (\frac{m−1}{2}) \sqrt{π}} ⟨ϕ, u⟩

= 2i^{j+2}j!T_{j}^{(a)}[(1 + 2a)^{−(j+1)}]\frac{Γ (\frac{m}{2})}{Γ (\frac{m−1}{2}) \sqrt{π}} ⟨ϕ, u⟩ϕ = −D_{m,j}⟨ϕ, u⟩ϕ.

where we used C_{0}ω_{m−1} = T_{0}^{(b)}[1] = (m − 2)^{−1} and (6.42) with k = j + 1 for T_{j}^{(a)}[(1+2a)^{−(j+1)}]. This proves statement (2). We omit the proof of statement (3) which is similar to the corresponding part of the previous lemma.

Lemma 6.10. Define D̃_{m} = ∑_{j=0}^{ν} D_{m,j}. Then, D̃_{m} = 1.

Proof. Use binomial formula for (6.53). We have

D̃_{m} = 2^{m} \frac{Γ (\frac{m}{2})}{Γ (\frac{m−1}{2}) \sqrt{π}} ∫_{1}^{∞} \frac{x^{m−2}}{(x^{2} + 1)^{m−1}} dx

Change of variable x → x^{−1} shows that the integral is equal to the same integral over the interval 0 < x < 1. It follows after making the change of variable x^{2} = t that the integral is equal to

\frac{1}{4} ∫_{0}^{∞} \frac{t^{ν− \frac{1}{2}}}{(t + 1)^{m−1}} dt = \frac{Γ (\frac{m−1}{2})^{2}}{2^{2}Γ(m − 1)}.

Thus, D̃_{m} = 2^{m−2}Γ (\frac{m}{2}) Γ (\frac{m−1}{2}) Γ(m − 1)^{−1}π^{− \frac{1}{2}} = 1.

In the next two sections we prove that Z^{νν} and Z_{log} are bounded in L^{p}(ℝ^{m}) for all 1 < p < ∞. These will complete the proof of Theorem 1.5.

6.2 Estimate of ∥Z^{νν}u∥_{p} for 1 < p < ∞

In this section we prove

∥Z^{νν}u∥_{p} ≤ C∥u∥_{p}, 1 < p < ∞.  (6.59)

The method of previous subsection does not apply for proving this and we exploit more direct method. By virtue of interpolation, it suffices to prove (6.59) for arbitrarily small p > 1 and large p > m.

6.2.1 The case for 1 < p < \frac{2(m−1)}{m+1}

We first show (6.59) for 1 < p < \frac{2(m−1)}{m+1}. After changing the variable r to (1 + 2a)r in (6.3), we write Q_{νν}^{a,b}(ρ)/ρ^{ν} in the form

\frac{(−1)^{ν+1}}{2πρ^{ν}} ∫_{0}^{∞} e^{i(1+2b)ρλ}λ^{m−3}F (λ) (∫_{ℝ} e^{−i(1+2a)rλ}r^{ν+1}M(r)dr) dλ. (6.60)

Integration by parts implies that (6.60) is equal to

\frac{i(−1)^{ν+1}}{2π(1 + 2b)ρ^{ν+1}} ∫_{0}^{∞} e^{i(1+2b)ρλ}(λ^{m−3}F (λ))′ (∫_{ℝ} e^{−i(1+2a)rλ}r^{ν+1}M(r)dr) dλ

+ \frac{(−1)^{ν+1}(1 + 2a)}{2π(1 + 2b)ρ^{ν+1}} ∫_{0}^{∞} e^{i(1+2b)ρλ}λ^{m−3}F (λ) (∫_{ℝ} e^{−i(1+2a)rλ}r^{ν+2}Mdr) dλ.

The first line becomes i(1 + 2b)^{−1}Q_{ν(ν−1)}^{a,b}(ρ)/ρ^{m−2−(ν−1)} if we replace (m − 3)F (λ) + λF′(λ) by F (λ) and the former function can play the same role as the latter does in the argument of previous sections and, ν − 1 ≥ 1 if m ≥ 6. Thus, if we substutute it for Q_{νν}^{a,b}(ρ)/ρ^{ν} in (6.8) for (j, k) = (ν, ν) and, then the resulting function for Z_{a,b}^{νν}(ϕ)u(x) in (6.7), it produces the operator which has the same L^{p} property as Z^{ν(ν−1)} which is bounded in L^{p}(ℝ^{m}) for 1 < p < ∞. Hence, we need study only the operator produced by the second line. Once again we substitute it for Q_{νν}^{a,b}(ρ)/ρ^{ν} in (6.8) and the result for Z_{a,b}^{νν}(ϕ)u(x) in (6.7). We denote the functiotn thus obtain by Z^{νν}u(x), abusing notation. We want to show that this Z^{νν}u(x) satisfies (6.59) for 1 < p < \frac{m}{m−1}. Integrating with respect to a, b first via Fubini’s theorem shows

Z^{νν}u(x) = \frac{2i}{ω_{m−1}} ∫_{ℝ^{m}}(V ϕ)(x − y)X_{ν}(|y|)dy,  (6.61)

X_{ν}(ρ) = \frac{2iC_{m,ν}^{2}ω_{m−1}}{ρ^{ν+1}} ∫_{0}^{∞} \{e^{iλρ}λ^{m−3} (∫_{0}^{∞} \frac{(1 + 2b)^{−1}e^{2iλρb}}{(1 + b)^{ν+ \frac{1}{2}}} \frac{db}{\sqrt{b}})

× ∫_{ℝ} e^{−iλr} (∫_{0}^{∞} \frac{(1 + 2a)e^{−2iaλr}}{(1 + a)^{ν+ \frac{1}{2}}} \frac{da}{\sqrt{a}}) r^{ν+2}M(r)dr\} F (λ)dλ.  (6.62)

Documenta Mathematica 21 (2016) 391–443

Let χ_{±}(r) = 1 for ±r > 0 and χ_{±}(r) = 0 for ±r ≤ 0. Define, for t > 0,

g_{±}(t) = ∫_{0}^{∞} (1 + \frac{a}{t}) (1 + \frac{a}{2t})^{−ν− \frac{1}{2}} e^{±ia} \frac{da}{\sqrt{a}},  (6.63)

h_{±}(t) = ∫_{0}^{∞} (1 + \frac{b}{t})^{−1} (1 + \frac{b}{2t})^{−ν− \frac{1}{2}} e^{±ib} \frac{db}{\sqrt{b}}  (6.64)

and, with C = iC_{m,ν}^{2}ω_{m−1}, write X_{ν}(ρ) as follows:

X_{ν}(ρ) = \frac{C}{ρ^{ν+ \frac{3}{2}}} ∫_{ℝ}(L_{+}(ρ, r) + L_{−}(ρ, r))r^{ν+2}|r|^{− \frac{1}{2}} M(r)dr,  (6.65)
L_{±}(ρ, r) = χ_{±}(r) ∫_{0}^{∞} e^{iλ(ρ−r)}λ^{m−4}h_{+}(λρ)g_{∓}(±rλ)F (λ)dλ.  (6.66)

Lemma 6.11. Suppose that f is C^{∞} on [0, ∞) and satisfies |f^{(j)}(c)| ≤ C_{j}c^{−(j+1)} for c ≥ 1, j = 0, 1, . . .. Define

ℓ_{±}(t) = ∫_{0}^{∞} e^{±ic}f(c/t)\frac{dc}{\sqrt{c}}.

Then, ℓ_{±}(t) is C^{∞} for t > 0 and satisfies the following properties.

(1) ℓ_{±}(1/t) can be exteded to a C^{∞} function on [0, 1], hence, lim_{t→∞} ℓ_{±}(t) = α_{±} exists and for t ≥ 1, |ℓ_{±}^{(j)}(t)| ≤ C_{j}t^{−j−1}, j = 1, 2, . . ..

(2) For 0 < t < 1, |t^{j}ℓ_{±}^{(j)}(t)| ≤ C_{j}\sqrt{t} ≤ C_{j}, j = 0, 1, . . ..

Proof. We prove the lemma for ℓ_{+}(t) only and omit the +-sign. It is evident that ℓ(t) is C^{∞} for t > 0. Splitting the interval, we define

ℓ(t) = (∫_{0}^{1} + ∫_{1}^{∞}) f (\frac{c}{t}) e^{ic} \frac{dc}{\sqrt{c}} ≡ ℓ_{1}(t) + ℓ_{2}(t).

It is obvious that ℓ_{1}(1/t) is of C^{∞}[0, 1]. To see the same for ℓ_{2}(1/t), we perform integration by parts n times for t > 0:

i^{n}ℓ_{2}(1/t) = B_{n}(t) + (−1)^{n} ∫_{1}^{∞} ∂_{c}^{n} (\frac{f(ct)}{\sqrt{c}}) e^{ic}dc.  (6.67)

The boundary term B_{n}(t) is a polynomial of order n and Leibniz’ formula implies ∂_{c}^{n} (\frac{f(ct)}{\sqrt{c}}) = ∑_{j=0}^{n} C_{nj}f^{(j)}(ct)(ct)^{j}c^{− \frac{1}{2} −n}. Since ∂_{y}^{k}(f^{(j)}(y)y^{j}) is bounded for any j, k = 0, 1, … and

∂_{t}^{k} (∑_{j=0}^{n} C_{nj}f^{(j)}(ct)(ct)^{j}c^{− \frac{1}{2} −n}) = ∑_{j=0}^{n} C_{nj} ∂_{y}^{k}(f^{(j)}(y)y^{j})|_{y=ct} c^{− \frac{1}{2} −n+k},

the integral of (6.67) is a function of class C^{n−1}([0, 1]). Since n is arbitray, this proves (1). For proving (2), after changing the variable we decompose:

ℓ(t) = \sqrt{t} (∫_{0}^{1} + ∫_{1}^{∞}) f(c)e^{ict} \frac{dc}{\sqrt{c}} ≡ \sqrt{t}(ℓ̃_{1}(t) + ℓ̃_{2}(t))

We obseve that \sqrt{t} satisfies the property (2) and that, if α(t) satisfies (2) and |t^{j}β^{(j)}(t)| ≤ C_{j}, then so does γ(t) = α(t)β(t). Hence, \sqrt{t}ℓ̃_{1}(t) satisfies (2) because ℓ̃_{1}(t) is entire. To prove the same for \sqrt{t}(ℓ̃_{2}(t), it suffices to show that |(t^{n}ℓ̃_{2}(t))^{(n)}| ≤ C_{n} for 0 < t < 1, n = 0, 1, 2, . . .. By integration by parts we have

(it)^{n}ℓ̃_{2}(t) = ∫_{1}^{∞}(∂_{c}^{n}e^{itc})f(c)\frac{dc}{\sqrt{c}}

= ∑_{j=0}^{n−1}(−1)^{j+1}∂_{c}^{j} (\frac{f(c)}{\sqrt{c}}) ∂_{c}^{n−j−1}(e^{itc})|_{c=1} + ∫_{1}^{∞} e^{itc}(f(c)c^{− \frac{1}{2}} )^{(n)}dc.

The boundary term is a polynomial of t and the integral is n times continuously differentiable and a fortiori (t^{n}ℓ̃_{2}(t))^{(n)} ≤ C for 0 < t < 1.

We define L_{±,σ}(ρ, r) for an integer σ ≥ 0 and functions g_{±} and h by

L_{±,σ}(ρ, r) = χ_{±}(r) ∫_{0}^{∞} e^{iλ(ρ−r)}λ^{σ}h_{+}(λρ)g_{∓}(±rλ)F (λ)dλ  (6.68)

so that we have L_{±}(ρ, r) = L_{±,m−4}(ρ, r) (see (6.66)).

Lemma 6.12. Suppose that g_{±}(t) and h_{+}(t) are C^{∞} functions of t > 0 and they satisfy following properties replacing f:

(a) The limit lim_{t→∞} f(t) exists.

(b) |t^{j}f^{(j)}(t)| ≤ C_{j} \{ \begin{matrix}t^{−1}, & 1 < t, & j = 1, 2, …, \\ \sqrt{t}, & 0 < t < 1, & j = 0, 1, …. \end{matrix} .

Then, L_{±,σ} is C^{∞} with respect to ρ > 0 and r > 0 and, for a constant C > 0,

|L_{±,σ}(ρ, r)| ≤ C⟨ρ − r⟩^{−(σ+1)}  (6.69)

Proof. We prove the lemma for L_{+,σ}. The proof for L_{−,σ} is similar. It is obvious that L_{+,σ}(ρ, r) is smooth and is bounded for ρ, r > 0 and, it suffices to prove (6.69) for |ρ − r| ≥ 1. We apply integration by parts σ + 1 times to

L_{+,σ}(ρ, r) = \frac{(−i)^{σ+1}}{(ρ − r)^{σ+1}} ∫_{0}^{∞} (∂_{λ}^{σ+1}e^{iλ(ρ−r)}) λ^{σ}h_{+}(λρ)g_{−}(rλ)F (λ)dλ.

By Leibniz’ rule, derivatives (λ^{σ}h_{+}(λρ)g_{−}(rλ)F (λ))^{(κ)} are linear combinations over indices (β, γ, δ) such that κ − σ ≤ β + γ + δ ≤ κ of

λ^{σ−κ+δ}(λρ)^{β}h^{(β)}(λρ)(rλ)^{γ}g_{−}^{(γ)}(rλ)F^{(δ)}(λ)  (6.70)

and they converge to 0 as λ → 0 if κ ≤ σ. It follows that (ρ−r)^{σ+1}L_{+,σ}(ρ, r) is the linear combination over the same set of (β, γ, δ) as above but with κ = σ+1 of

I_{βγδ}(ρ, r) = ∫_{0}^{∞} e^{i(ρ−r)λ}λ^{δ−1}(λρ)^{β}h^{(β)}(λρ)(rλ)^{γ}g_{−}^{(γ)}(rλ)F^{(δ)}(λ)dλ.

It suffices to show that I_{βγδ}(ρ, r) is bounded. If δ ≠ 0, F^{(δ)}(λ) = 0 outside 0 < c_{0} < λ < c_{1} < ∞ and it is clear that I_{βγδ}(ρ, r)≤_{| · |} C. Thus, we assume δ = 0 in what follows. We may also assume 0 < r < ρ < ∞ by symmetry. We split the interval of integration as (0, ∞) = (0, 1/ρ) ∪ [1/ρ, 1/r] ∪ (1/r, ∞) and denote integrals over these intervals by I_{1}, I_{2} and I_{3} in this order so that I_{βγδ}(ρ, r) = I_{1} + I_{2} + I_{3}.

(1) If 0 < λ < 1/ρ then 0 < rλ < ρλ < 1 and (ρλ)^{β}h^{(β)}(ρλ)≤_{| · |} C\sqrt{ρλ} and (rλ)^{γ}g_{−}^{(γ)}(rλ)≤_{| · |} C\sqrt{rλ}. It follows that

I_{1}≤_{| · |} C ∫_{0}^{1/ρ} \sqrt{ρr}dλ = C\sqrt{\frac{r}{ρ}} ≤ C  (6.71)

(2) If 1/ρ ≤ λ ≤ 1/r, we have 0 < rλ ≤ 1 ≤ ρλ and we estimate as (ρλ)^{β}h^{(β)}(ρλ)≤_{| · |} C and (rλ)^{γ}g_{−}^{(γ)}(rλ)≤_{| · |} C\sqrt{rλ}. It follows that

I_{2}≤_{| · |} C ∫_{1/ρ}^{1/r} λ^{− \frac{1}{2}} \sqrt{r}dλ = 2C\sqrt{r} (\frac{1}{\sqrt{r}} − \frac{1}{\sqrt{ρ}}) ≤ 2C.  (6.72)

(3) Finally if 1 < rλ < ρλ, then we likewise estimate

(λρ)^{β}h^{(β)}(λρ)(rλ)^{γ}g_{−}^{(γ)}(rλ)≤_{| · |} C \{ \begin{matrix}(rλ)^{−1}, & if β = 0, γ ≠ 0 \\ (ρλ)^{−1}, & if β ≠ 0, γ = 0, \\ (ρλ)^{−1}(rλ)^{−1}, & if β, γ ≠ 0. \end{matrix}

The right hand side is bounded by Cr^{−1}λ^{−1} and

I_{3}≤_{| · |} C ∫_{1/r}^{∞} λ^{−2}r^{−1}dλ = C.

This completes the proof.

Proposition 6.13. Let m ≥ 6 and ϕ ∈ E. For 1 ≤ p ≤ \frac{2(m−1)}{m+1}, we have

∥Z^{νν}u∥_{p} ≤ C_{p}∥u∥_{p}.  (6.73)

Proof. We recall (6.61). Lemma 6.12 implies L_{±}(ρ, r)≤_{| · |} C⟨ρ − r⟩^{−(m−3)}. It follows by Young’s inequality and (6.65) that

∥Z^{νν}u∥_{p} ≤ C∥V ϕ∥_{1} (∫_{0}^{∞} (∫_{ℝ} \frac{ρ^{\frac{m−1}{p} − \frac{m+1}{2}} |r^{\frac{m+1}{2}} M(r)|}{⟨ρ − r⟩^{m−3}}dr)^{p} dρ)^{\frac{1}{p}} . (6.74)

Documenta Mathematica 21 (2016) 391–443

Define κ = \frac{m−1}{p} − \frac{m+1}{2}, then κ ≥ 0 for 1 ≤ p ≤ \frac{2(m−1)}{m+1} and m − 3 − κ ≥ \frac{3}{2} for any 1 ≤ p < ∞ if m ≥ 6. Thus, we may estimate

ρ^{κ}⟨ρ − r⟩^{−(m−3)} ≤ C \{ \begin{matrix}⟨ρ − r⟩^{− \frac{3}{2}} & if |r| ≤ 1 \\ ⟨ρ − r⟩^{− \frac{3}{2}} |r^{κ}| & if |r| ≥ 1 \end{matrix}

and Young’s inequality implies

∥Z^{νν}u∥_{p} ≤ C∥V ϕ∥_{1} (∫_{0}^{1} |r^{\frac{m+1}{2}} M(r)|^{p}dr + ∫_{1}^{∞} |r^{\frac{m−1}{p}} M(r)|^{p}dr)^{\frac{1}{p}} ,

which is bounded by C(∥V ϕ∗u∥_{∞} +∥V ϕ∗u∥_{p}) ≤ (∥V ϕ∥_{p′} +∥V ϕ∥_{1})∥u∥_{p}. This completes the proof of the proposition.

6.2.2 The case \frac{2(m−1)}{m−3} ≤ p < ∞

Lemma 6.14. Let m ≥ 6 and ϕ ∈ E. Then, Z^{νν}(ϕ) is bounded in L^{p}(ℝ^{m}) for any \frac{2(m−1)}{m−3} ≤ p < ∞.

Proof. we apply integration by parts to (6.60) by using the identity that ∫_{ℝ} e^{−i(1+2a)rλ}r^{ν+1}M(r)dr = i(1 + 2a)^{−1}∂_{λ} (∫_{ℝ} e^{−i(1+2a)rλ}r^{ν}M(r)dr)′. We see that ρ^{−ν}Q_{νν}^{a,b}(ρ) is equal to

\frac{(−1)^{ν+1}}{2πρ^{ν}(i(1 + 2a))} ∫_{0}^{∞} e^{i(1+2b)ρλ}(λ^{m−3}F (λ))′ (∫_{ℝ} e^{−i(1+2a)rλ}r^{ν}M(r)dr) dλ

+ \frac{(−1)^{ν+1}(1 + 2b)}{2πρ^{ν−1}(1 + 2a)} ∫_{0}^{∞} e^{i(1+2b)ρλ}λ^{m−3}F (λ) (∫_{ℝ} e^{−i(1+2a)rλ}r^{ν}M(r)dr) dλ

The argument similar to the one at the beginning of the proof of Proposition 6.13 shows that the operator produced by the first line has the same L^{p} property as Z^{(ν−1)ν} and, hence, is bounded in L^{p}(ℝ^{m}) for any 1 < p < ∞. Thus, we need consider the operator produced by the second line, which we substitute for Q_{νν}^{a,b}(ρ)/ρ^{ν} in (6.8) and the resulting function for Z_{a,b}^{νν}(ϕ)u(x) in (6.7). The result is given by (6.61) where X_{ν}(ρ) is replaced by

X̃_{ν}(ρ) = \frac{C}{ρ^{ν−1}} ∫_{0}^{∞} \{e^{iλρ}λ^{m−3} (∫_{0}^{∞} \frac{(1 + 2b)e^{2iλρb}}{(1 + b)^{ν+ \frac{1}{2}}} \frac{db}{\sqrt{b}})

× ∫_{ℝ} e^{−iλr} (∫_{0}^{∞} \frac{(1 + 2a)^{−1}e^{−2iaλr}}{(1 + a)^{ν+ \frac{1}{2}}} \frac{da}{\sqrt{a}}) r^{ν}M(r)dr\} F (λ)dλ,  (6.75)

which can be simplified into the form (6.65), (6.66) with the roles of g and h being replaced and the factors ρ^{−(ν+ \frac{3}{2})} and r^{ν+2}|r|^{− \frac{1}{2}} being replaced by ρ^{−(ν− \frac{1}{2})} and r^{ν}|r|^{− \frac{1}{2}} respectively. Then, Lemmas 6.11 and 6.12 imply

X_{ν}(ρ)≤_{| · |} \frac{C}{ρ^{ν− \frac{1}{2}}} ∫_{ℝ}⟨ρ − r⟩^{3−m}|r|^{ν}|r|^{− \frac{1}{2}} M(r)dr.

We estimate ∥X_{ν}(|y|)∥_{L^{p}(|y|≥1)} for p ≥ \frac{2(m−1)}{m−3} and ∥X_{ν}(|y|)∥_{L^{1}(|y|<1)}. Let κ = \frac{m−1}{p} − ν + \frac{1}{2}. If p ≥ \frac{2(m−1)}{m−3}, then κ ≤ 0 and m − 3 + κ ≥ \frac{3}{2} for m ≥ 6 and for ρ ≥ 1

ρ^{κ}⟨ρ − r⟩^{3−m}|r|^{ν− \frac{1}{2}} ≤ C⟨ρ − r⟩^{− \frac{3}{2}} ⟨r⟩^{κ}|r|^{ν− \frac{1}{2}} ≤ C⟨ρ − r⟩^{− \frac{3}{2}} |r|^{\frac{m−1}{p}} .

It follows by Young’s inequality that for any 2 ≤ p < ∞,

∥X_{ν}(|y|)∥_{L^{p}(|y|≥1)} ≤ C (∫_{0}^{∞} |∫_{ℝ}⟨ρ − r⟩^{− \frac{3}{2}} |r|^{\frac{m−1}{p}} M(r)dr|^{p} dρ)^{\frac{1}{p}}

≤ C (∫_{0}^{∞} |M(r)|^{p}r^{m−1}dr)^{\frac{1}{p}} ≤ C∥V ϕ∥_{1}∥u∥_{p}. (6.76)

When ρ ≤ 1, we have ρ^{m−1−ν+ \frac{3}{2}} ≤ 1 and ⟨ρ − r⟩^{3−m} ≤ C⟨r⟩^{3−m}. Hence,

∥X_{ν}(|y|)∥_{L^{1}(|y|<1)} ≤ C ∫_{ℝ}⟨r⟩^{3−m}|r|^{ν− \frac{1}{2}} |M(r)|dr ≤ C∥M∥_{∞} ≤ C∥V ϕ∥_{p′}∥u∥_{p}.

We therefore obtain by using Young’s inequality again after splitting the integral corresponding to (6.61) into the ones over |y| < 1 and |y| ≥ 1 that

∥Z^{νν}u∥_{p} ≤ C(∥V ϕ∥_{1}^{2} + ∥V ϕ∥_{p}∥V ϕ∥_{p′})∥u∥_{p}.

This completes the proof.

6.3 Estimate of ∥Z_{log}u∥_{p}

In this section we study Z_{log} and prove the following lemma. The combination of the lemma with results of the previous subsections proves Theorem 1.5 for even dimensions m ≥ 6, the formal proof of which will be omitted.

Lemma 6.15. (1) If m = 6, then Z_{log} is bounded in L^{p}(ℝ^{m}) for all 1 < p < m. If E = E_{0}, then so is Z_{log} for all 1 < p < ∞.

(2) If m ≥ 8, then Z_{log} is bounded in L^{p}(ℝ^{m}) for all 1 < p < ∞.

Proof. We prove the lemma for m = 6 only. The proof for m ≥ 8 is similar and easier. Out of three operators on the right of (3.27) for m = 6, we first study

Z_{1,log} = ∫_{0}^{∞} G_{0}(λ)(V φ ⊗ V φ)λ log λ(G_{0}(λ) − G_{0}(−λ))F (λ)dλ,  (6.77)

where we have ignored the constant ω_{m−1}/π(2π)^{m} which is not important. Since Z_{1,log} = 0, if E = E_{0}, it suffices to prove (1) for 1 < p < \frac{m}{m−1} and \frac{m}{2} < p < m. By using (2.6) and (2.18) as previously, we express Z_{1,log} as the sum over 0 ≤ j, k ≤ ν of

Z_{1,log}^{jk}u(x) = C_{jk}T_{j}^{(a)}T_{k}^{(b)} [∫_{ℝ^{m}} \frac{(V ϕ)(x − y)Q_{jk,log}^{a,b}(|y|)}{|y|^{m−2−k}}dy] ,  (6.78)

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where Q_{jk,log}^{a,b}(ρ) are defined by (6.3) or (6.5) (for the case j = 0) by replacing λ^{j+k−1} or λ^{k} respectively by λ^{j+k+1} log λ or λ^{k+2} log λ. We prove

∥Z_{1,log}^{jk}u∥_{p} ≤ C∥u∥_{p}  (6.79)

separately for (j, k) ≠ (ν, ν) and (j, k) = (ν, ν) by repeating the argument in corresponding subsections. Let (j, k) ≠ (ν, ν). We first observe that, if j ≥ 1, Fourier inverse transforms of the derivatives upto the order k + 1 of λ^{j+k+1}(log λ)F have the RDIM

F^{∗}(λ^{j+k+1}(log λ)F)^{(l)})(ρ)≤_{| · |} C(1 + ρ)^{−2}⟨log(1 + ρ)⟩, 0 ≤ l ≤ k + 1

and estimates corresponding to (6.11) and (6.27) are satisfied by Q_{jk,log}^{a,b}(ρ) respectively for 1 ≤ j ≤ ν and for j = 0 (without producing the boundary term). Then, the argument in §6.1.1 goes through for Z_{1,log}^{jk} and produces estimate (6.79) for 1 < p < \frac{m}{m−1}. By the same reason the estimate corresponding (6.13) for m/2 < p < m is satisfied by Q_{jk,log}^{a,b}(ρ) for all j, k and we likewise have (6.79) for m/2 < p < m by using the argument of the first part of proof of Lemma 6.8. It is then obvious that the same holds for Z_{2,log} which is obtained from Z_{1,log} by replacing λ log λ by λ^{3}(log λ) and, that the operator

Z_{3,log}^{(a,b)} = ∫_{0}^{∞} G_{0}(λ)(φ_{a} ⊗ ψ_{b})λ^{3}(log λ)^{2}(G_{0}(λ) − G_{0}(−λ))F (λ)dλ. (6.80)

produced by λ^{2} log λF_{2} of (3.18) satisfies (6.79) for all 1 < p < m. We next prove (6.79) when (j, k) = (ν, ν). It suffices prove it for 1 < p < p_{0} for some p_{0} > 1 and p ≥ p_{1} for some p_{1} > 2. The argument at the beginnings of §6.2.1 and §6.2.2 shows that respectively for 1 < p < p_{0} and p ≥ p_{1}, we have only to estimate operators obtained by replacing Q_{jk,log}^{a,b}(ρ) by

\frac{1 + 2a}{(1 + 2b)ρ^{ν+1}} ∫_{0}^{∞} e^{i(1+2b)ρλ}λ^{m−1} log λF (λ) (∫_{ℝ} e^{−i(1+2a)rλ}r^{ν+2}Mdr) dλ

and

\frac{1 + 2b}{(1 + 2a)ρ^{ν−1}} ∫_{0}^{∞} e^{i(1+2b)ρλ}λ^{m−1} log λF (λ) (∫_{ℝ} e^{−i(1+2a)rλ}r^{ν+2}Mdr) dλ

in (6.78). We then repeat the argument of §6.2. We have λ^{m−2} log λ in place of λ^{m−4} in (6.66). If we change λ^{σ} by λ^{σ+2} log λ in the definition (6.68) of L̃_{±}(ρ, r), then (6.69) is satisfied with faster decaying factor ⟨ρ − r⟩^{−(σ+2)} in place of ⟨ρ − r⟩^{−(σ+1)}. Thus, ∥Z_{log}^{νν}u∥_{p} is bounded C∥V ϕ∥_{1} times (6.74) with ⟨ρ−r⟩^{−(m−2)} in place of ⟨ρ−r⟩^{−(m−3)} and this proves the lemma for 1 < p < p_{0}. The proof for p ≥ p_{1} is similar and we omit further details.

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K. Yajima Department of Mathematics Gakushuin University 1-5-1 Mejiro Toshima-ku Tokyo 171-8588 Japan 444

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