WEIGHTED NORM INEQUALITIES FOR

INTEGRAL OPERATORS AND RELATED TOPICS

Vladimir D. Stepanov

Introduction

In the first part of the paper we study integral operators of the form

(1)  Kf(x) = v(x) ∫_{0}^{x} k(x, y)u(y)f(y) dy, x > 0,

where the real weight functions v(t) and u(t) are locally integrable and the kernel k(x, y) ≥ 0 satisfies the following condition: there exists a constant D ≥ 1 such that

D^{−1}(k(x, y) + k(y, z)) ≤ k(x, z) ≤ D(k(x, y) + k(y, z)), (2)

x > y > z ≥ 0,

where D does not depend on x, y, z. The condition (2) was introduced by R. Oinarov [O_{1}].

Standard examples of a kernel k(x, y) ≥ 0 satisfying (2) are

(i) k(x, y) = (x − y)^{α}, α ≥ 0

(ii) k(x, y) = log^{β}(1 + x − y), k(x, y) = log^{β} (\frac{x}{y}); β ≥ 0

and their various combinations.

Let 0 < p ≤ ∞. We study (1) on ℝ^{+} = (0, ∞), but any (a, b) ⊂ ℝ can be taken instead of (0, ∞) without any loss of generality. Also, the dual operator of T can be considered (see Remark 1.1 below). Let

This research was partially supported by NSERC of Canada and SERC of United Kingdom.

L^{p} = L^{p}(0, ∞) = {f: ∥f∥_{p} = ( ∫_{0}^{∞}|f(x)|^{p}dx)^{1/p} < ∞}.

We consider K as a map from L^{p} into L^{q} and shall characterize the following problems:

(B) L^{p} − L^{q} boundedness,

(C) L^{p} − L^{q} compactness and measure of non-compactness,

(S) L^{2} − L^{2} Schatten-von Neumann ideal norms.

Several factors affect the problem (B). First of these are restrictions imposed on the kernel k(x, y) ≥ 0. Another such factor is the range of parameters p and q, because of substantial difference between the cases p ≤ q and q < p. Certain part is played also by the fact whether 1 ≤ p, q ≤ ∞ or not. The cases when p = 1 or p = ∞; q ≥ 1 and q = 1 or q = ∞; p ≥ 1 follow trivially from known results ([KA], Chapter XI, Theorem 4). It also follows from the general theory of integral operators [AS], [Sc] that if 0 < p < 1 and K: L^{p} → L^{q} is bounded, then k(x, y) = 0 almost everywhere. Among other factors, perhaps, the verifiability of a criterion is the most relevant. For instance, Muckenhoupt’s criterion [M] for the L^{p} − L^{p} boundedness of (1) when k(x, y) = 1 penetrated many areas because of its explicit form, and being so easy to verify. On the other hand, the implicit “Schur’s test” [Kor], [Sz], given for arbitrary kernel k(x, y) ≥ 0, 1 < q ≤ p < ∞, had also had effective applications [Nik], [Hern].

The problem (B) was intensively studied since 1988, when the characterization has been found for the Riemann–Liouville operator with kernel k(x, y) = (x − y)^{α}, α ≥ 0 and its convolution generalizations [MS] (1 < p ≤ q < ∞), [St_{1}] (1 < p, q < ∞), [St_{2}–St_{6}], [St_{8}], [St_{10}]. Unlike the Muckenhoupt (1 < p ≤ q < ∞) or Mazja (1 < q < p < ∞) criteria for the case k(x, y) = 1 (see [B], [Ko], [M], [Ma], [Tal], [Tom], [Saw_{1}], [S_{1}] for the case 0 < q < 1 < p < ∞, and [OK] for the full account), the L^{p} − L^{q} boundedness of the Riemann–Liouville operator was characterized by two conditions, which are independent in general except the case when u(y) = 1 or v(x) = 1. Later, in the papers [BK_{1}] and [O_{2}] (1 < p ≤ q < ∞), [St_{12}] (1 < p, q < ∞), the criteria have been proved for the kernels satisfying monotonicity or continuity conditions. The classical results with the power weights can be found in [HLP].

The problem (C) has the background in the spectral theory of weighted string [G], [AO], the case k(x, y) = 1 was proved in [Riem], and for the Riemann–Liouville operator see [St_{2}–St_{3}].

In Section 1.1 we characterize the problems (B) and (C) for the Volterra integral operators (1) under the condition (2), which probably is a balance point between generality of conditions imposed on a kernel and implicitness of a criterion. A few extensions to Lorentz and Orlicz spaces are given in Sections 1.2 and 1.3 which generalize the results of [AM], [EGP], [Saw_{1}] and [BK_{2}], [HM], [L], respectively.

The problem (S) is quite a natural step from the problems (B) and (C). The standard references in this area are the books [GK], [K], [P_{1}], [Sim] and the survey article [BS]. Applying the real interpolation method [BL] we give in the Section 1.4 an analog of the Hilbert–Schmidt formulae for the Schatten-von Neumann ideal σ_{p}-norm, 2 ≤ p < ∞, for operators of the form (1) with extension to the range 1 ≤ p < ∞ for the polynomial kernel.

In the second part we deal with weighted norm inequalities restricted to monotone functions. This topic was recently initiated by the papers [ArM] and [Saw_{2}] with further developments in [A], [Br], [CS_{1}], [CS_{2}], [G_{1}–G_{3}], [H], [HSt], [N], [St_{7}], [St_{9}], [St_{11}] and others.

Our first observation is that such inequalities are helpful in the above problem (B) for the Hardy operator, when 0 < q < p < ∞, p ≥ 1.

Secondly, we characterize the dual space to the classical Lorentz space given by

Γ_{p}(v) = \{f: ∥f∥_{p,v} = ( ∫_{0}^{∞}(\frac{1}{t} ∫_{0}^{t} f^{∗}(s) ds)^{p}v(t) dt)^{1/p} < ∞\},

where f^{∗}(s) = inf[z ≥ 0: meas {x: |f(x)| > z} ≤ s], and find out the criteria for a number of operators of harmonic analysis to be bounded in Γ-spaces.

Throughout the paper the expressions of the form 0 · ∞, 0/0, ∞/∞ are taken equal to zero, the inequality A ≪ B means A ≤ cB, where c depends only on D and parameters of summation (p, q …), and the relationship A ≈ B is interpreted as A ≪ B ≪ A or A = cB. Further, χ_{E} denotes the characteristic function of a set E, ℤ the set of all integers, □ means the end of a proof.

Acknowledgement. It is a pleasure of the author to express his deep respect and gratitude to Professors M. Krbec, A. Kufner, B. Opic and J.

Rákosník, the organizers of the School on Nonlinear Analysis, Function Spaces and Applications, for their kind invitation and hospitality. He also wishes to thank the International Soros Foundation for the financial support for his participation at the School.

1. Problems (B) and (C)

1.1 Lebesgue spaces. Denote \frac{1}{p} + \frac{1}{p′} = 1, \frac{1}{q} + \frac{1}{q′} = 1, \frac{1}{q} − \frac{1}{p} = \frac{1}{r}, and

A_{0} = sup_{t>0} A_{0}(t) = sup_{t>0} ( ∫_{t}^{∞}k^{q}(x, t)|v(x)|^{q}dx)^{1/q}( ∫_{0}^{t} |u(y)|^{p′}dy)^{1/p′}, (3)

A_{1} = sup_{t>0} A_{1}(t) = sup_{t>0} ( ∫_{t}^{∞}|v(x)|^{q}dx)^{1/q}( ∫_{0}^{t} k^{p′}(t, y)|u(y)|^{p′}dy)^{1/p′}, (4)

B_{0} = { ∫_{0}^{∞}( ∫_{t}^{∞}k^{q}(x, t)|v(x)|^{q}dx)^{r/q}( ∫_{0}^{t} |u(y)|^{p′}dy)^{r/q′}|u(t)|^{p′}dt}^{1/r}, (5)

B_{1} = { ∫_{0}^{∞}( ∫_{t}^{∞}|v(x)|^{q}dx)^{r/q}( ∫_{0}^{t} k^{p′}(t, y)|u(y)|^{p′}dy)^{r/p′}|v(t)|^{q}dt}^{1/r}, (6)

The mapping properties of K: L^{p} → L^{q} for 1 ≤ p, q ≤ ∞ (with the usual modification if p = 1, ∞ or q = 1, ∞) are described by the following statement.

Theorem 1.1. Let the integral operator K be given by (1) with the kernel k(x, y) ≥ 0 satisfying (2).

(a_{1}) If 1 < p ≤ q < ∞ then the inequality

(7)  ∥Kf∥_{q} ≤ C∥f∥_{p} for all f ∈ L^{p}

is valid if, and only if, A = max(A_{0}, A_{1}) < ∞ and, moreover, ∥K∥ ≈ A.

(a_{2}) If 1 < p ≤ q < ∞ then the operator K: L^{p} → L^{q} is compact if, and only if, A < ∞ and

(8)  lim_{t→0} A_{i}(t) = lim_{t→∞} A_{i}(t) = 0, i = 0, 1.

(b_{1}) If 1 < q < p < ∞ then the inequality (7) holds if, and only if, B = max(B_{0}, B_{1}) < ∞ and, moreover, ∥K∥ ≈ B.

(b_{2}) If 1 < q < p < ∞ then the operator K: L^{p} → L^{q} is compact if, and only if, B < ∞.

Proof. (a_{1}) We begin with the necessity. Observe that the left hand side of (2) implies

(9) k(t, y) ≤ Dk(x, y) and k(x, t) ≤ Dk(x, y) for all y < t < x.

Now, suppose that (7) and the inequality

(10)  ∫_{0}^{t} k^{p′}(t, y)|u(y)|^{p′}dy < ∞

hold for some t > 0 (otherwise we would have ∫_{t}^{∞} |v|^{q} = 0 and the convention 0 · ∞ = 0 would imply 0 = A_{1}(t) ≤ C). Setting

f_{t}(x) = χ_{[0,t]}(x)(k(t, x)|u(x)|)^{p′−1} sgn u(x),

then substituting f_{t}(x) in (7) and applying (9), we find that

C( ∫_{0}^{t} k^{p′}(t, y)|u(y)|^{p′}dy)^{1/p}

≥ ( ∫_{0}^{∞}|v(x)|^{q}dx( ∫_{0}^{x} k(x, y)χ_{[0,t]}(y)(k(t, y))^{p′−1}|u(y)|^{p′}dy)^{q} dx)^{1/q}

≥ D^{−1}( ∫_{t}^{∞}|v(x)|^{q}dx)^{1/q}( ∫_{0}^{t} k^{p′}(t, y)|u(y)|^{p′}dy)

and the estimate D^{−1}A_{1} ≤ C follows.

By duality, we have

K: L^{p} → L^{q} ⇔ K^{∗}: L^{q′} → L^{p′},

where

K^{∗}g(y) = u(y) ∫_{y}^{∞}k(x, y)v(x)g(x) dx.

Thus (7) and ∥K^{∗}g∥_{p′} ≤ C∥g∥_{q′}, g ∈ L^{q′}, hold with the same constant C. Applying the above argument to the operator K^{∗} and

g_{t}(y) = χ_{[t,∞]}(y)(k(y, t)|v(y)|)^{q−1} sgn v(y),

we see that D^{−1}A_{0} ≤ C, D^{−1}A ≤ C, and D^{−1}A ≤ inf C = ∥K∥.

To prove sufficiency we need the following Lemma 1.1. Let f(y) be such that u(y)f(y) ≥ 0 and k(x, y) ≥ 0 satisfy (2). Put

G(x) = ∫_{0}^{x} k(x, y)u(y)f(y) dy,

and

X_{k} = {x > 0: G(x) ≥ (δ + 1)^{k}}, k ∈ ℤ,

x_{k} = inf X_{k} if X_{k} ≠ ∅, x_{k} = ∞ otherwise; N = sup{k: X_{k} ≠ ∅},

where δ > 0 is a fixed number. If δ > D^{3}, then the inequality

(δ + 1)^{k−1} ≤ ∫_{x_{k−1}}^{x_{k}} k(x_{k}, y)u(y)f(y) dy + D ∫_{x_{k−2}}^{x_{k−1}}k(x_{k−1}, y)u(y)f(y) dy

+ Dk(x_{k}, x_{k−1}) ∫_{0}^{x_{k−1}}u(y)f(y) dy

+ D^{2}l(x_{k−1}, x_{k−2}) ∫_{0}^{x_{k−2}}u(y)f(y) dy

holds.

Proof. By the definition we have x_{k−1} ≤ x_{k} and also, G(x) ≤ (δ + 1)^{k} ≤ G(x_{k}), if x ∈ [x_{k−1}, x_{k}). Using this, we write

(δ + 1)^{k+1} = (δ + 1)^{2}((δ + 1)^{k} − δ(δ + 1)^{k−1})

≤ (δ + 1)^{2}(G(x_{k}) − δ(δ + 1)^{k−1})

= (δ + 1)^{2}( ∫_{0}^{x_{k}}k(x_{k}, y)u(y)f(y) dy − δ(δ + 1)^{k−1}).

Now applying (2) twice we get

(δ + 1)^{k−1} ≤Dk(x_{k}, x_{k−1}) ∫_{0}^{x_{k−1}}u(y)f(y) dy + ∫_{x_{k−1}}^{x_{k}} k(x_{k}, y)u(y)f(y) dy

+ D ∫_{0}^{x_{k−1}}k(x_{k−1}, y)u(y)f(y) dy − δ(δ + 1)^{k−1}

≤Dk(x_{k}, x_{k−1}) ∫_{0}^{x_{k−1}}u(y)f(y) dy + ∫_{x_{k−1}}^{x_{k}} k(x_{k}, y)u(y)f(y) dy

+ D^{2}k(x_{k−1}, x_{k−2}) ∫_{0}^{x_{k−2}}u(y)f(y) dy

+ D ∫_{x_{k−2}}^{x_{k−1}}k(x_{k−1}, y)u(y)f(y) dy

+ D^{2}G(x_{k−2}) − δ(δ + 1)^{k−1}.

Using (9) we find that G(x_{k−2}) ≤ DG(x) ≤ D(δ + 1)^{k−1} if x_{k−2} ≤ x ≤ x_{k−1}. Consequently, D^{2}G(x_{k−2}) < δ(δ + 1)^{k−1}, if D^{3} < δ, and the result follows. □

Now we continue with the proof of the sufficiency part of Theorem 1.1.

Without loss of generality we may and shall assume that f has a compact support in ℝ^{+}, f(y)u(y) ≥ 0 and 0 < ∥f∥_{p} < ∞. By Lemma 1.1 we obtain

J = ∫_{0}^{∞}G^{q}|v|^{q} = ∑_{k∈N} ∫_{x_{k}}^{x_{k+1}}G^{q}|v|^{q} ≤ ∑_{k≤N}(δ + 1)^{q(k+1)} ∫_{x_{k}}^{x_{k+1}}|v|^{q} (11)

≤4^{q−1}(δ + 1)^{2q} ∑_{k} ∫_{x_{k}}^{x_{k−1}}|v|^{q}{D^{q}k^{q}(x_{k}, x_{k−1})( ∫_{0}^{x_{k−1}}uf)^{q}

+ ( ∫_{x_{k−1}}^{x_{k}} k(x_{k}, y)u(y)f(y) dy)^{q} + D^{2q}k^{q}(x_{k−1}, x_{k−2})( ∫_{0}^{x_{k−2}}uf)^{q}

+ D^{q}( ∫_{x_{k−2}}^{x_{k−1}}k(x_{k−1}, y)u(y)f(y) dy)^{q}}

= 4^{q−1}(δ + 1)^{2q}(J_{11} + J_{12} + J_{21} + J_{22}).

Using the Hölder and the Jensen inequalities, we find

J_{12} = ∑_{k} ∫_{x_{k}}^{x_{k+1}}|v|^{q}( ∫_{x_{k−1}}^{x_{k}} k(x_{k}, y)u(y)f(y) dy)^{q}

≤ ∑_{k} ∫_{x_{k}}^{x_{k+1}}|v|^{q}( ∫_{x_{k−1}}^{x_{k}} k^{p′}(x_{k}, y)|u(y)|^{p′}dy)^{q/p′}( ∫_{x_{k−1}}^{x_{k}} |f(y)|^{p}dy)^{q/p}

≤ A_{1}^{q} ∑_{k} ( ∫_{x_{k−1}}^{x_{k}} |f(y)|^{p}dy)^{q/p} ≤ A_{1}^{q}∥f∥_{p}^{q}.

Analogously, we get

J_{22} = ∑_{k} ∫_{x_{k}}^{x_{k+1}}|v|^{q}( ∫_{x_{k−2}}^{x_{k−1}}k^{p′}(x_{k−1}, y)|u(y)|^{p′}dy)^{q} ≤ A_{1}^{q}D^{q}∥f∥_{p}^{q}.

In case when k(x, y) ≡ 1, the single condition A_{1} < ∞ is necessary and sufficient for (7), and A_{0} ≈ A_{1}. To obtain upper bounds for J_{11} and J_{21} we need the following more general assertion.

Lemma 1.2 ([BK_{2}]). Let 1 < p ≤ q < ∞ and let y = ϕ(x) be a differentiable increasing function on ℝ^{+} such that ϕ(0) = 0, ϕ(∞) = ∞ and, thus, the inverse function x = ϕ^{−1}(y) exists. Then

∥v(x) ∫_{0}^{ϕ(x)}fu∥_{q} ≤ C∥f∥_{p} for all f ∈ L^{p},

if, and only if,

A_{ϕ} = sup_{t>0} ∥χ_{[ϕ^{−1}(t),∞]}v∥_{q}∥χ_{[0,t]}u∥_{p′} < ∞,

and, moreover, C ≈ A_{ϕ}.

Proof follows from the case k(x, y) ≡ 1 by a change of variables. □

We can now estimate J_{11} and J_{21} as follows. We have

J_{11} = D^{q} ∑_{k} ( ∫_{x_{k}}^{x_{k+1}}|v|^{q})k^{q}(x_{k}, x_{k−1})( ∫_{0}^{x_{k−1}}u(y)f(y) dy)^{q}.

Put Φ(x) = ∑_{k} x_{k−2}χ_{[x_{k},x_{k+1}]}(x) and let y = ϕ(x) be such a function that

ϕ(x_{k}) = x_{k−2}, Φ(x) ≤ ϕ(x) ≤ x, and ϕ(x) satisfies the hypothesis of Lemma 1.2, that is, ϕ is an increasing C^{1} function on (0, ∞). Then we obtain

J_{11} ≪ D^{q} ∑_{k} ( ∫_{x_{k}}^{x_{k+1}}|v|^{q})k^{q}(x_{k}, x_{k−1})[( ∫_{x_{k−2}}^{x_{k−1}}uf)^{q} + ( ∫_{0}^{x_{k−2}}uf)^{q}]

= D^{q}(J_{11}^{(1)} + J_{11}^{(2)}).

The estimate of J_{11}^{(1)} is similar to J_{12}, but now we apply Hölder’s inequality and (9) to get

J_{11}^{(1)} ≤ D^{q} ∑_{k} ( ∫_{x_{k}}^{x_{k+1}}k^{q}(x, x_{k−1})|v(x)|^{q}dx)( ∫_{x_{k−2}}^{x_{k−1}}uf)^{q}

≤ D^{q} ∑_{k} ( ∫_{x_{k}}^{∞}k^{q}(x, x_{k−1})|v(x)|^{q}dx)( ∫_{x_{k−2}}^{x_{k−1}}|u|^{p′})^{q/p′}( ∫_{x_{k−2}}^{x_{k−1}}|f|^{q})^{q/p}

≤ A_{1}^{q}D^{q} ∑_{k} ( ∫_{x_{k−2}}^{x_{k−1}}|f|^{p})^{q/p} ≤ A_{1}^{q}D^{q}∥f∥_{p}^{q}.

For the second term we obtain from Lemma 1.2

J_{11}^{(2)} = ∑_{k} ( ∫_{x_{k}}^{x_{k+1}}|v(x)|^{q}dx)k^{q}(x_{k}, x_{k−1})) ∫_{0}^{x_{k−2}}uf)^{q}

≤ ∫_{0}^{∞}( ∫_{0}^{Φ(x)}uf)^{q}( ∑_{k} k^{q}(x_{k}, x_{k−1})χ_{[x_{k},x_{k−1}]}(x))|v(x)|^{q}dx

≤ ∫_{0}^{∞}( ∫_{0}^{ϕ(x)}uf)^{q}V (x) dx ≪ (A_{ϕ}^{(1)})^{q}∥f∥_{p}^{q},

where

V (x) = ( ∑_{k} k^{q}(x_{k}, x_{k−1})χ_{[x_{k},x_{k−1}]}(x))|v(x)|^{q}

and

A_{ϕ}^{(1)} = sup_{t>0} A_{ϕ}^{(1)}(t), A_{ϕ}^{(1)}(t) = ( ∫_{ϕ^{−1}(t)}^{∞} V (x) dx)^{1/q}( ∫_{0}^{t} |u|^{p′})^{1/p′}.

Now by the definition of ϕ(x) we have that if ϕ^{−1}(t) ∈ [x_{k_{0}}, x_{k_{0}+1}), then t ∈ [x_{k_{0}−2}, x_{k_{0}−1}) and, in particular, t ≤ ϕ^{−1}(t). Applying (9) twice we find that

(12) ∫_{ϕ^{−1}(t)}^{∞} V (x) dx = k^{q}(x_{k_{0}}, x_{k_{0}−1}) ∫_{ϕ^{−1}(t)}^{x_{k_{0}+1}}|v(x)|^{q}dx

+ ∑_{k>k_{0}} k^{q}(x_{k_{0}}, x_{k_{0}−1}) ∫_{x_{k}}^{x_{k+1}}|v(x)|^{q}dx

≤D^{2q}( ∫_{t}^{x_{k_{0}+1}}k^{q}(x, t)|v(x)|^{q}dx + ∑_{k>k_{0}} ∫_{x_{k}}^{x_{k+1}}k^{q}(x, t)|v(x)|^{q}dx)

=D^{2q} ∫_{t}^{∞}k^{q}(x, t)|v(x)|^{q}dx.

Hence, A_{ϕ}^{(1)} ≤ D^{2}A_{0}, and we obtain J_{11} ≪ A^{q}∥f∥_{p}^{q}. A similar argument shows that J_{21} ≪ A^{q}∥f∥_{p}^{q}, and the estimate J ≪ A^{q}∥f∥_{p}^{q} is proved.

Remark 1.1. Observe that the part (a_{1}) has two natural versions:

(i) restricted to any interval of real line and (ii) with respect to the dual operator K^{∗}. In the case (i), which we call a “restricted” version, we deal with the integral operator K given by

Kf(x) = v(x) ∫_{a}^{x} k(x, y)u(y)f(y) dy, −∞ ≤ a < x < b ≤ ∞,

with the kernel k(x, y) ≥ 0 satisfying

D^{−1}(k(x, y) + k(y, z)) ≤ k(x, z) ≤ D(k(x, y) + k(y, z)), (2’)

b ≥ x < y < z ≥ a.

Then ∥K∥_{L^{p}(a,b)→L^{q}(a,b)} ≈ A_{a,b} for 1 < p ≤ q < ∞, where A_{a,b} = max(A_{0;a,b}, A_{1;a,b}), and

A_{0;a,b} = sup_{a<t<b} A_{0;a,b}(t) = sup_{t>0} ( ∫_{t}^{b} k^{q}(x, t)|v(x)|^{q}dx)^{1/q}( ∫_{a}^{t} |u(y)|^{p′}dy)^{1/p′},

A_{1;a,b} = sup_{a<t<b} A_{1;a,b}(t) = sup_{t>0} ( ∫_{t}^{b} |v(x)|^{q}dx)^{1/q}( ∫_{a}^{t} k^{p′}(t, y)|u(y)|^{p′}dy)^{1/p′}.

Analogously, in the “dual” version (ii) we have ∥K^{∗}∥_{L^{p}(a,b)→L^{q}(a,b)} ≈ A_{(a,b)}^{∗}, where A_{a,b}^{∗} = max(A_{0;a,b}, A_{1;a,b}), and

A_{0;a,b}^{∗} = sup_{a<t<b} A_{0;a,b}^{∗}(t) = sup_{t>0} ( ∫_{a}^{t} k^{q}(t, x)|v(x)|^{q}dx)^{1/q}( ∫_{t}^{b} |u(y)|^{p′}dy)^{1/p′},

A_{1;a,b}^{∗} = sup_{a<t<b} A_{1;a,b}^{∗}(t) = sup_{t>0} ( ∫_{a}^{t} |v(x)|^{q}dx)^{1/q}( ∫_{t}^{b} k^{p′}(t, y)|u(y)|^{p′}dy)^{1/p′}.

(a_{2}) To prove necessity of (8) we use the well-known fact that a compact operator maps a weakly convergent sequence into a strongly convergent one. As before we may and shall assume (10) for some t > 0 and, consequently, for all s < t. Put

f_{s}(x) = \frac{χ_{[0,s]}(x)(k(s, x)|u(x))^{p′−1} sgn u(x)}{( ∫_{0}^{s} k^{p′}(s, y)|u(y)|^{p′}dy)^{1/p}}, 0 < s < t.

Then for any fixed g ∈ L^{p′} we have by Hölder’s inequality that

| ∫_{0}^{∞}f_{s}(x)g(x) dx| ≤ ( ∫_{0}^{s} |g(x)|^{p′}dx)^{1/p′} → 0, s → 0.

Hence, f_{s} → 0 is a weakly convergent sequence, and, by the hypotheses, we have

lim_{s→0} ∥Kf_{s}∥_{q} = 0.

However, (9) yields

∥Kf_{s}∥_{q} = ( ∫_{0}^{∞}|v(x)|^{q}( ∫_{0}^{x} k(x, y)f_{s}(y)u(y) dy)^{q}dx)^{1/q}

≥ D^{−1}( ∫_{t}^{∞}|v(x)|^{q}dx)^{1/q}( ∫_{0}^{s} k^{p′}(s, y)|u(y)|^{p′}dy)^{1/p′}.

Consequently, lim_{s→0} A_{1}(t) = 0. Using the same argument with the sequence

f_{s}(x) = \frac{χ_{[0,s]}(x)|u(x)|^{p′−1} sgn u(x)}{( ∫_{0}^{s} |u(y)|^{p′}dy)^{1/p}}, 0 < s < t.

we obtain lim_{s→0} A_{0}(t) = 0. The second part of (8) follows on applying similar observations with respect to the dual operator which is compact, too.

For the proof of sufficiency observe that if A < ∞ and there exists t > 0 such that

∫_{0}^{t} k^{p′}(t, y)|u(y)|^{p′}dy = ∞,

then

∫_{t}^{∞}|v(x)|^{q}dx = 0

and we have to restrict our considerations to the interval [0, t]. Together with the similar observation at the other end of the real semiaxis this shows that without loss of generality we may and shall assume that all the factors in A_{i}(t), i = 0, 1 are finite when 0 < t < ∞.

Let 0 < a < b < ∞ and

P_{a}f = χ_{[0,a]}f, Q_{b}f = χ_{[x,∞]}f, P_{ab}f = χ_{[a,b]}f.

Then we have

Kf = (P_{a} + P_{ab} + Q_{b})K(P_{a} + P_{ab} + Q_{b})f
= P_{a}KP_{a}f + Q_{b}KQ_{b}f + P_{ab}KP_{ab}f + Q_{b}KP_{ab}f + Q_{a}KP_{a}f.

By (a_{1}) restricted to the intervals [0, a] or [b, ∞], and (8), we have

∥P_{a}KP_{a}∥ ≤ max{ sup_{0<t<a} A_{0}(t), sup_{0<t<a} A_{1}(t)} → 0, a → 0,
∥Q_{b}KQ_{b}∥ ≤ max{sup_{t>b} A_{0}(t), sup_{t>b} A_{1}(t)} → 0, b → ∞.

To finish the proof of (a_{2}) we need the following

Lemma 1.3 ([ES_{1}], Lemma 1). Let 1 < p ≤ q < ∞ and 0 < a < b < ∞, let K be an operator of the form (1) with a kernel k(x, y) ≥ 0 satisfying (2). Then if A < ∞, the maps P_{ab}KP_{ab}, Q_{b}KP_{ab} and Q_{a}KP_{a} are compact.

Thus, K is compact as a limit of compact operators.

(b_{1}) We begin with the necessity part. Let 1 < q < p < ∞, 1/r = 1/q − 1/p, and let (7) hold. Put

V_{0}(t) = ∫_{t}^{∞}|v(x)|^{q}dx,  V_{k}(t) = ∫_{t}^{∞}k^{q}(x, t)|v(x)|^{q}dx,

U_{0}(t) = ∫_{0}^{t} |u(y)|^{p′}dy,  U_{k}(t) = ∫_{0}^{t} k^{p′}(t, y)|u(y)|^{p′}dy,

and assume that B_{0} < ∞. To get this inequality we can proceed by changing the weight functions u and v without changing C.

We set

f_{k} = (V_{k}^{1/q}U_{0}^{1/q′})^{r/p}|u|^{p′/p} sgn u.

152 V.D. STEPANOV

Then B_{0} = ∥f_{k}∥_{p}^{p/r}, and replacing in (7) f by f_{k}, we get C∥f_{k}∥_{p} = CB_{0}^{r/p} ≥ ∥Kf_{k}∥_{q}

= ( ∫_{0}^{∞}|v(t)|^{q}dt ∫_{0}^{t} k(t, y)u(y)f_{k}(y) dy( ∫_{0}^{t} k(t, x)u(x)f_{k}(x) dx)^{q−1})^{1/q}

= ( ∫_{0}^{∞}f_{k}(y)u(y) dy ∫_{y}^{∞}k(t, y)|v(t)|^{q}dt( ∫_{0}^{t} k(t, x)u(x)f_{k}(x) dx)^{q−1})^{1/q}

(by (9))

≥ D^{−1/q′}( ∫_{0}^{∞}f_{k}(y)u(y) dy ∫_{y}^{∞}k^{q}(t, y)|v(t)|^{q}dt( ∫_{0}^{y} u(x)f_{k}(x) dx)^{q−1})^{1/q}

(applying (9) again we find that V_{k}(x) ≥ D^{−q}V_{k}(y) if 0 < x < y)

≥ D^{−1}( ∫_{0}^{∞}f_{k}(y)u(y)(V_{k}(y))^{(p−1)/(p−q)}dy( ∫_{0}^{y} |u(x)|^{p′}U_{0}(x) dx)^{q−1})^{1/q}

≈ B_{0}^{r/q}. The above estimate gives

C ≥ (\frac{p − q}{p − 1})^{1/q′}D^{−1}B_{0},

and the temporary assumption B_{0} < ∞ can be removed thanks to the Fatou lemma. A similar argument applied to the dual operator gives

C ≥ (\frac{p − q}{p − 1})^{1/p}D^{−1}B_{1},

and the required lower bound follows.

The proof of the sufficiency part begins exactly in the same way as the proof of (a_{1}). Preserving the notation (11) we write

J ≡ ∥Kf∥_{q}^{q} ≪ (J_{11} + J_{12} + J_{21} + J_{22}).

Applying twice Hölder’s inequality and then Jensen’s inequality we find

WEIGHTED NORM INEQUALITIES 153

that

J_{12} = ∑_{k} ∫_{x_{k}}^{x_{k+1}}|v|^{q}( ∫_{x_{k−1}}^{x_{k}} k(x_{k}, y)u(y)f(y) dy)^{q}

≤ ∑_{k} ∫_{x_{k}}^{x_{k+1}}|v|^{q}( ∫_{x_{k−1}}^{x_{k}} k^{p′}(x_{k}, y)|u(y)|^{p′}dy)^{q/p′}( ∫_{x_{k−1}}^{x_{k}} |f(y)|^{p}dy)^{q/p}

≤ ( ∑_{k} ( ∫_{x_{k}}^{x_{k+1}}|v|^{q})^{r/q}( ∫_{x_{k−1}}^{x_{k}} k^{p′}(x_{k}, y)|u(y)|^{p′}dy)^{r/p′})^{q/r}( ∑_{k} ∫_{x_{k−1}}^{x_{k}} |f(y)|^{p}dy)^{q/p}

≤ \frac{r}{q} ( ∑_{k} ∫_{x_{k}}^{x_{k+1}}( ∫_{x}^{∞}|v|^{q})^{r/p}|v(x)|^{q}dx( ∫_{x_{k−1}}^{x_{k}} k^{p′}(x_{k}, y)|u(y)|^{p′}dy)^{r/p′})^{q/r}∥f∥_{p}^{q}

≤ \frac{r}{q} D^{q}( ∑_{k} ∫_{x_{k}}^{x_{k+1}}( ∫_{0}^{x} k^{p′}(x, y)|u(y)|^{p′}dy)^{r/p′}( ∫_{x}^{∞}|v|^{q})^{r/q}|v(x)|^{q}dx)^{q/r}∥f∥_{p}^{q}

≤ \frac{r}{q} B_{1}^{q}D^{q}∥f∥_{p}^{q}.

Similarly we get

J_{22} ≤ \frac{r}{q} B_{1}^{q}D^{q}∥f∥_{p}^{q}.

As in the proof of (a_{1}) it is clear at the moment that in the case k(x, y) = 1, the single condition B_{1} < ∞ is necessary and sufficient for (7), and B_{0} ≪ ∥K∥ ≪ B_{1}. Let us show that B_{1} ≪ B_{0} in this case. This is obvious if B_{0} = ∞, therefore suppose B_{0} < ∞. Then

( ∫_{t}^{∞}|v|^{q})^{r/q}( ∫_{0}^{t} |u|^{p′})^{r/p′} = ( ∫_{t}^{∞}|v|^{q})^{r/q} ∫_{0}^{t} d( ∫_{0}^{s} |u|^{p′})^{r/p′}

≤ ∫_{0}^{t} ( ∫_{t}^{∞}|v|^{q})^{r/q}d( ∫_{0}^{s} |u|^{p′})^{r/p′}

= \frac{r}{p′} ∫_{0}^{t} ( ∫_{s}^{∞}|v|^{q})^{r/q}( ∫_{0}^{s} |u|^{p′})^{r/q′}|u(s)|^{p′}ds

≤ \frac{r}{p′} B_{0} < ∞,

and, consequently,

( ∫_{t}^{∞}|v|^{q})^{r/q}( ∫_{0}^{t} |u|^{p′})^{r/p′} → 0 for t → 0

by the Lebesgue Dominated Convergence Theorem. Hence, integrating by parts, we see that

B_{0}^{r} = ∫_{0}^{∞}( ∫_{t}^{∞}|v|^{q})^{r/q}( ∫_{0}^{t} |u|^{p′})^{r/p′}|u(t)|^{p′}dt

= \frac{p′}{r} ∫_{0}^{∞}( ∫_{t}^{∞}|v|^{q})^{r/q}d( ∫_{0}^{t} |u|^{p′})^{r/p′}

≤ \frac{p′}{r} ∫_{0}^{∞}( ∫_{t}^{∞}|v|^{q})^{r/p}( ∫_{0}^{t} |u|^{p′})^{r/p′}|v(t)|^{q}dt = \frac{p′}{r} B_{1}^{r}.

Also, as in the proof of (a_{1}), we need the slightly extended version of this case which can be proved easily by the change of variables.

Lemma 1.4. Let 1 < q < p < ∞ and let y = ϕ(x) be a differentiable increasing function on ℝ^{+}, such that ϕ(0) = 0, ϕ(∞) = ∞, and thus the inverse function x = ϕ^{−1}(y) exists. Then

∥v(x) ∫_{0}^{ϕ(x)}fu∥_{q} ≤ C∥f∥_{p} for all f ∈ L^{p},

if, and only if,

B_{ϕ} = ( ∫_{0}^{∞}( ∫_{ϕ^{−1}(t)}^{∞} |v|^{q})^{r/p}( ∫_{0}^{t} |u|^{p′})^{r/p′}|u(t)|^{p′}dt)^{1/r} < ∞,

and, moreover, C ≈ B_{ϕ}.

Now we continue with the proof of sufficiency in (b_{1}) by applying the construction from (a_{1}), involving the functions Φ(x) and ϕ(x) which have been used for the upper bound of J_{11} and J_{21}. As before we have

J_{11} ≪ D^{q}(J_{11}^{(1)} + J_{11}^{(2)}).

The estimate of J_{11}^{(1)} is similar to that of J_{12}, so we have J_{11}^{(1)} ≪ B_{1}^{q}D^{q}∥f∥_{p}^{q}. Applying Lemma 1.4 for the upper bound of J_{11}^{(2)} we find J_{11}^{(2)} ≪ (B_{ϕ}^{(1)})^{q}∥f∥_{p}^{q}, where

B_{ϕ}^{(1)}(t) = ( ∫_{0}^{∞}( ∫_{ϕ^{−1}(t)}^{∞} V (x) dx)^{r/q}( ∫_{0}^{t} |u|^{p′})^{r/q′}|u(t)|^{p′}dt)^{1/r},

and using (12) we obtain J_{11}^{(2)} ≪ B_{0}^{q}∥f∥_{p}^{q}. The upper bound of J_{21} follows by a similar argument. Thus, combining the above estimates, we get J ≪ B^{q}∥f∥_{p}^{q} and the part (b_{1}) is established.

(b_{2}) Necessity follows immediately from (b_{1}), and the Ando theorem [An] implies sufficiency.

Theorem 1.1 is proved. □

Remark 1.2. The measure of non-compactness of K: L^{p} → L^{q} is given by

α(K) = inf ∥K − P ∥,

where the infimum is taken over all bounded linear maps P: L^{p} → L^{q} of finite rank. Using the restricted version of Theorem 1.1 we can show that in the case 1 < p ≤ q < ∞ we have

α(K) ≈ max(A_{L}, A_{R}),

where A_{L} = lim_{a→0} ∥P_{a}KP_{a}∥; A_{R} = lim_{b→∞} ∥P_{b}KP_{b}∥. (See [ES_{1}] for details.)

1.2. Lorentz spaces. For 0 < r < ∞, 0 < s ≤ ∞, and a locally integrable function ϕ(x) on ℝ^{+}, the Lorentz space L_{ϕ}^{rs} ≡ L_{ϕ}^{rs}(ℝ^{+}) consists of all measurable functions f such that ∥f∥_{rs,ϕ} < ∞, where

∥f∥_{rs,ϕ} = ( ∫_{0}^{∞}(t^{1/r}f^{∗∗}(t))^{s} \frac{dt}{t})^{1/s} for 0 < s < ∞,
∥f∥_{r∞,ϕ} = sup_{t>0} t^{1/r}f^{∗∗}(t) for s = ∞,

and

f^{∗∗}(t) = ∫_{0}^{t} f^{∗}(s) ds,

f^{∗}(t) = inf {x > 0: λ_{f}(x) =  ∫_{{y∈ℝ^{+} : |f(y)|>x}} ϕ(z) dz ≤ t}.

If r = s, then

∥f∥_{rr,ϕ} = ( ∫_{0}^{∞}|f(x)|^{r}ϕ(x) dx)^{1/r}.

There is a natural extension of Theorem 1.1, (a), to Lorentz spaces. We set

A_{0} = sup_{t>0} A_{0}(t) = sup_{t>0} ∥χ_{[0,t]}(.)k(t, .)(u(.)/ϕ(.))∥_{r′s′,ϕ}∥χ_{[t,∞]}v∥_{pq,ψ},
A_{1} = sup_{t>0} A_{1}(t) = sup_{t>0} ∥χ_{[0,t]}(u/ϕ)∥_{r′s′,ϕ}∥χ_{[t,∞]}(.)k(., t)v(.)∥_{pq,ψ},

Theorem 1.2 ([LS]). Let K be the integral operator (1) with a kernel k(x, y) ≥ 0 satisfying (2), and let 1 < r, p < ∞, 1 ≤ s, q ≤ ∞ be such that

(13)  max(r, s) ≤ min(p, q).

(a_{1}) Then

(14)  ∥Kf∥_{pq,ψ} ≤ C∥f∥_{rs,ϕ}, f ∈ L_{ϕ}^{rs}

if, and only if,

A = max(A_{0}, A_{1}) < ∞

and, moreover, ∥K∥ ≈ A, where ∥K∥ is the norm of K, i.e., the least possible constant C in (14).

(a_{2}) If 1 < r, p < ∞, 1 ≤ s, q < ∞ and (13) holds, then K: L_{ϕ}^{rs} → L_{ψ}^{pq} is compact if, and only if, A < ∞ and

lim_{t→0} A_{i}(t) = lim_{t→∞} A_{i}(t) = 0, i = 0, 1 ….

Proof of Theorem 1.2 can be obtained by applying the scheme of the proof of Theorem 1.1 and Lemma 3 from [EGP], which is a substitute for Jensen’s inequality in the Lorentz spaces. □

Remark 1.3. (a) The case k(x, y) = 1 of Theorem 1.2 has recently been proved in [EGP], Theorem 3.

(b) The results on the measure of non-compactness, analogous to those mentioned in Remark 1.2, are valid for Lorentz spaces. Moreover, using the “restricted” version of Theorem 1.2 and schemes from [EEH] and [ES_{1}] it is possible to obtain upper and lower bounds of the approximation numbers of K (see [LS] for details).

(c) In Section 2.1 below a criterion is given for the boundedness of the Hardy operator in Lorentz spaces in the case when (13) is not satisfied.

1.3. Orlicz spaces. Let Φ: ℝ^{+} → ℝ^{+} be a nonnegative, convex function such that

lim_{x→0} \frac{Φ(x)}{x} = 0, lim_{x→∞} \frac{Φ(x)}{x} = ∞.

Following [KR] we call Φ an N-function. Several authors ([BK_{2}], [HM], [L]) have recently established criteria for the validity of the inequality

Φ_{2}^{−1}( ∫_{0}^{∞}Φ_{2}(|Kf(x)|)|ω(x)| dx) (15)

≤Φ_{1}^{−1}( ∫_{0}^{∞}Φ_{1}(C|f(x)|)|ρ(x)| dx),

where ω(x) and ρ(x) are locally integrable, Φ_{1}, Φ_{2} are two N-functions satisfying appropriate conditions, and K is an integral operator of type (1) with a kernel monotone with respect to x and y. Using the arguments from [BK_{2}], [L] and Lemma 1.1 we obtain the following

Theorem 1.3. Let K be the integral operator (1) with a kernel k(x, y) ≥ 0 satisfying (2), let Φ_{1}, Φ_{2} be two N-functions with complementary N-functions Ψ_{1}, Ψ_{2}, respectively, and such that Φ_{2} ∘ Φ_{1}^{−1} is convex. Then (15) holds if, and only if, 𝔸 < ∞, where 𝔸 = 𝔸_{0} + 𝔸_{1} and

𝔸_{0} = inf \{τ > 0: sup_{t>0} sup_{λ>0} ∫_{0}^{t} Ψ_{1}(\frac{α(λ, t)k(t, x)|u(x)|}{τλ|ρ(x)|})\frac{|ρ(x)|}{α(λ, t)} dx ≤ 1\},

𝔸_{1} = inf \{τ > 0: sup_{t>0} sup_{λ>0} ∫_{0}^{t} Ψ_{1}(\frac{β(λ, t)|u(x)|}{τλ|ρ(x)|})\frac{|ρ(x)|}{β(λ, t)} dx ≤ 1\}

with

α(λ, t) = Φ_{1} ∘ Φ_{2}^{−1}( ∫_{t}^{∞}Φ_{2}(λ|v(x)|)|ω(x)| dx),

and

β(λ, t) = Φ_{1} ∘ Φ_{2}^{−1}( ∫_{t}^{∞}Φ_{2}(λk(x, t)|v(x))|ω(x)| dx).

Moreover, the best possible constant from (15) satisfies C ≈ 𝔸.

Remark 1.4. The statement of Theorem 1.3 is essentially taken from [BK_{2}], Theorem 1.7. An alternative version is given in [L], Theorem 1, which also has an extension for the kernels satisfying (2) without monotonicity conditions (we omit the details). Both papers as well as [O_{2}] and the present one make use of the Martín–Reyes and Sawyer method ([MS]), which is applicable to the non-monotone kernels in view of the technical Lemma 1.1, which, in particular, allows to resist the temptation to reduce the problem to a continuous kernel.

1.4 Schatten-von Neumann ideal norms. Let H be a separable Hilbert space. Then the set of all linear bounded operators T: H → H forms the normed algebra B, where σ_{∞} is the ideal of all compact operators. The theory of symmetrically normed (s.n.) ideals σ_{Φ} ⊂ σ_{∞}was developed by using the s.n. functions Φ defined on the space of sequences with a finite number of non-zero terms ([GK], Chapter 3). If T ∈ σ_{∞}, then T^{∗} ∈ σ_{∞} and (T^{∗}T)^{1/2} ∈ σ_{∞}. To construct σ_{∞} the sequences of singular numbers s_{j}(T) = λ_{j}[(T^{∗}T)^{1/2}] were used with eigenvalues λ_{j} ≥ 0 taken with respect to their multiplicity and in a decreasing ordering. The formula ∥T ∥_{σ_{Φ}} = Φ(s_{j}(T)) defines the norm (quasinorm) in the s.n. ideal σ_{Φ}. The most wellknown s.n. ideals are those, related to the space of sequences l_{p}, 0 < p ≤ ∞, and called σ_{p}. The norm (quasinorm) ∥T ∥ = ( ∑_{j} s_{j}^{p}(T))^{1/p} is usually called the Schatten-von Neumann norm (quasinorm). Thus, ∥T ∥_{σ_{∞}} = ∥T ∥ and ∥T ∥_{σ_{2}} is the Hilbert-Schmidt norm defined for an integral operator T f(x) = ∫ T (x, y)f(y) dy by the formula ∥T ∥_{σ_{2}} = ( ∫∫ |T (x, y)|^{2}dx dy)^{1/2}. It is known [BS], that in general the norm ∥T ∥_{σ_{p}} of an integral operator essentially depends on the smoothness of its kernel when p < 2. The aim of this section is to present a brief account of some results from [ES_{2}] about the Schatten-von Neumann ideal norms for the integral operator (1) under the condition (2) for its kernel.

Let H = L^{2}(0, ∞) and

A_{0}^{2} = sup_{t>0} ∫_{t}^{∞}k^{2}(x, t)|v(x)|^{2}dx ∫_{0}^{t} |u(y)|^{2}dy,

A_{1}^{2} = sup_{t>0} ∫_{t}^{∞}|v(x)^{2}dx ∫_{0}^{t} k^{2}(t, y)|u(y)|^{2}dy.

Theorem 1.1 and the Hilbert-Schmidt formula yield

∥K∥_{σ_{∞}} ≈ A_{0} + A_{1},
∥K∥_{σ_{2}} = ( ∫_{0}^{∞}|v(x)|^{2}dx ∫_{0}^{x} k^{2}(t, y)|u(y)|^{2}dy)^{1/2}

= ( ∫_{0}^{∞}|u(y)|^{2}dy ∫_{y}^{∞}k^{2}(x, y)|v(x)|^{2}dx)^{1/2}.

Using these formulas and applying the real interpolation method we obtain the following

Theorem 1.4. Let K be an operator of the form (1) with a kernel satisfying (2) and K ∈ σ_{∞}. Then

∥K∥_{σ_{p}} ≈( ∫_{0}^{∞}[( ∫_{0}^{x} k^{2}(x, y)|u(y)|^{2}dy)^{p/2}( ∫_{x}^{∞}|v(y)|^{2}dy)^{p/2−1}|v(x)|^{2} (16)

+ ( ∫_{x}^{∞}k^{2}(y, x)|v(y)|^{2}dy)^{p/2}( ∫_{0}^{x} |u(y)|^{2}dy)^{p/2−1}|u(x)|^{2}] dx)^{1/p},

2 ≤ p < ∞.

Remark 1.5. In the case k(x, y) ≡ 1 the formula (16) can be simplified as follows. If

Hf(x) = v(x) ∫_{0}^{x} f(y)u(y) dy,

then

(17) \frac{1}{2}∥H∥_{σ_{p}} ≤ ( ∫_{0}^{∞}( ∫_{0}^{x} |u(y)|^{2}dy)^{p/2}( ∫_{x}^{∞}|v(y)|^{2}dy)^{\frac{p}{2} −1}|v(x)|^{2}dx)^{1/p}
≤ ∥H∥_{σ_{p}}, 2 ≤ p < ∞.

For the values 1 ≤ p < 2 we obtain the following necessary conditions, using the approach of the recent paper by K. Nowak [Now].

Theorem 1.5. Let 1 ≤ p < ∞ and let the integral operator K ∈ σ_{p} be given by (1) with a kernel k(x, y) ≥ 0 satisfying (2). Then

∥K∥_{σ_{p}}

≤ (2D)^{−1} sup_{{a_{m}}} { ∑_{m} [( ∫_{a_{m−1}}^{a_{m}} k^{2}(a_{mk}, y)|u(y)|^{2}dy)^{p/2}( ∫_{a_{m}}^{a_{m+1}}|v(x)|^{2}dx)^{p/2}

+ ( ∫_{a_{m−1}}^{a_{m}} |u(y)|^{2}dy)^{p/2}( ∫_{a_{m}}^{a_{m+1}}k^{2}(x, a_{m})|v(x)|^{2}dx)^{p/2}]}^{1/p},

where the supremum is taken over all sequences 0 < ⋯ < a_{m−1} < a_{m} < a_{m+1} < . . ..

Corollary 1.1. In addition to (17) we have

∥H∥_{σ_{p}} ≥ p2^{−p−1}( ∫_{0}^{∞}( ∫_{0}^{x} |u(y)|^{2}dy)^{p/2}( ∫_{x}^{∞}|v(y)|^{2}dy)^{\frac{p}{2} −1}|v(x)|^{2}dx)^{1/p},
1 ≤ p < 2.

Now we restrict our consideration to the convolution operators with polynomial kernels of the form

T f(x) = v(x) ∫_{0}^{x} P_{n}(x − y)u(y)f(y) dy,

where

P_{n}(x) = α_{n}x^{n} + ⋯ + α_{1}x + α_{0}, α_{n} ≠ 0

is a polynomial with real coefficients. Using the scheme developed in [ES_{1}], we get an upper bound for ∥T ∥_{σ_{p}}. Denote |P_{n}|(x) = |α_{n}|x^{n} + ⋯ + |α_{1}|x + |α_{0}|.

Theorem 1.6. Let 0 < p < ∞. Then

∥T ∥_{σ_{p}}

≪ sup_{{a_{m}}} { ∑_{m} [( ∫_{a_{m−1}}^{a_{m}} |P_{n}|^{2}(a_{m} − y)|u(y)|^{2}dy)^{p/2}( ∫_{a_{m}}^{a_{m+1}}|v(x)|^{2}dx)^{p/2}

+ ( ∫_{a_{m−1}}^{a_{m}} |u(y)|^{2}dy)^{p/2}( ∫_{a_{m}}^{a_{m+1}}|P_{n}|^{2}(x − a_{m})|v(x)|^{2}dx)^{p/2}]}^{1/p}.

When α_{n} = 1, α_{n−1} = 0, …, α_{0} = 0, so that

T_{n}f(x) = v(x) ∫_{0}^{x}(x − y)^{n}u(y)f(y) dy, n ≥ 0,

we obtain the following supplement to the results of Theorems 1.4 and 1.5.

Corollary 1.2. Let 1 ≤ p < ∞ and let n ≥ 0 be an integer. Then

∥T_{n}∥_{σ_{p}} ≈ sup_{{a_{m}}} { ∑_{m} [( ∫_{a_{m−1}}^{a_{m}} (a_{m} − y)^{2n}|u(y)|^{2}dy)^{p/2}( ∫_{a_{m}}^{a_{m+1}}|v(x)|^{2}dx)^{p/2}

+( ∫_{a_{m−1}}^{a_{m}} |u(y)|^{2}dy)^{p/2}( ∫_{a_{m}}^{a_{m+1}}(x − a_{m})^{2n}|v(x)|^{2}dx)^{p/2}]}^{1/p}.

Remark 1.6. An alternative method to obtain the estimates of the Schattenvon Neumann norms for T_{n}, n > \frac{1}{2}, in the case when u(y) = 1, is given in [NS].

2. Weighted inequalities on the cones of monotone functions

2.1. Let 0 < p, q < ∞, and let v(x) ≥ 0 and u(x) ≥ 0 be locally integrable weight functions on ℝ^{+}, let f ↓ denote a non-negative non-increasing function on ℝ^{+}, and let the similar notation f ↑ stand for a non-decreasing function. Put

V (t) = ∫_{0}^{t} v, W (t) = ∫_{0}^{t} w.

Our first result is the following

Theorem 2.1. (a) Let 0 < p ≤ q < ∞. Then

(18)  sup_{f↓} \frac{( ∫_{0}^{∞}f^{q}w)^{1/q}}{( ∫_{0}^{∞}f^{p}v)^{1/p}} ≈ sup_{t>0} W^{1/q}(t)V^{1/p}(t) ≡ A.

(b) Let 0 < q < p < ∞, 1/r = 1/q − 1/p. Then

(19)  sup_{f↓} \frac{( ∫_{0}^{∞}f^{q}w)^{1/q}}{( ∫_{0}^{∞}f^{p}v)^{1/p}} ≈ B ≈ B + \frac{W^{1/q}(∞)}{V^{1/p}(∞)},

where

B = ( ∫_{0}^{∞}W^{r/p}V^{−r/p}w)^{1/r}, B = ( ∫_{0}^{∞}W^{r/q}V^{−r/q}v)^{1/r}.

(c) The same results hold for non-decreasing functions.

Of special interest is the particular case q = 1, which provides the principle of duality, that is, reverse Hölder inequality for monotone functions.

Corollary 2.1. (a) Let 1 < p < ∞, 1/p + 1/p′ = 1, and let g(x) ≥ 0 be locally integrable. Then

(20) sup_{f↓} \frac{∫_{0}^{∞}fg}{( ∫_{0}^{∞}f^{p}v)^{1/p}} ≈ ( ∫_{0}^{∞}( ∫_{0}^{s} g)^{p′−1}V^{1−p′}(s)g(s) ds)

≈ ( ∫_{0}^{∞}( ∫_{0}^{s} g)^{p′}V^{−p′}(s)v(s) ds)^{1/p′} + \frac{∫_{0}^{∞}g}{( ∫_{0}^{∞}v)^{1/p}} .

(b) If 0 < p ≤ 1, then

(21)  sup_{f↓} \frac{∫_{0}^{∞}fg}{( ∫_{0}^{∞}f^{p}v)^{1/p}} ≈ sup_{t>0} V^{−1/p}(t) ∫_{0}^{t} g.

Remark 2.1. The original proof of (20) is due to E. Sawyer ([Saw_{2}], Theorem 1), as well as Theorem 2.1 for the range 1 < p, q < ∞, which was established in [Saw_{2}] in terms of embeddings of classical Lorentz spaces. In full scope one can find the proof of Theorem 2.1 in [St_{9}], Proposition 1. Also, (21) was independently proved in [CS_{1}].

Corollary 2.1 is useful in many ways. In particular, it allows to reduce the weighted inequalities for monotone functions to those for arbitrary functions. Another, rather curious application is that, conversely, ordinary inequalities can be proved by using (20), too. The following theorem demonstrates this fact. Recall that the Lorentz space L_{ϕ}^{rs} ≡ L_{ϕ}^{rs}(ℝ^{+}) (see Section 1.2) is equipped with an equivalent quasinorm [SW]

∥f∥_{rs,ϕ}^{∗} = ( ∫_{0}^{∞}(t^{1/r}f^{∗}(t))^{s} \frac{dt}{t})^{1/s}, 0 < r, s < ∞.

Theorem 2.2. Let 0 < q < r < ∞, r ≥ 1, 0 < p < ∞, 1/γ = 1/q − 1/r,

and let the operator H be given by Hf(x) = ∫_{0}^{x} f(y)u(y) dy. Then

sup_{f≠0} \frac{∥Hf∥_{pq,ψ}^{∗}}{∥f∥_{rr,ϕ}^{∗}} ≈ ( ∫_{0}^{∞}(∥χ_{[t,∞]}∥_{pq,ψ}^{∗})^{γ}d(∥χ_{[0,t]}(u/ϕ)∥_{r′r′,ϕ}^{∗})^{γ})^{1/γ} ≡ B_{0}.

Proof. We begin with the sufficiency part. Suppose that f(x) ≥ 0 is a function with a compact support and such that 0 < ∥f∥_{rr,ϕ}^{∗} < ∞. We have

∥Hf∥_{pq,ψ}^{∗} = (\frac{1}{q} ∫_{0}^{∞}( ∫_{t}^{∞}ψ)^{q/p}d(Hf(t))^{q})^{1/q}.

Applying (20) in the form

∫_{0}^{∞}G dμ ≪ ( ∫_{0}^{∞}G^{s}dυ)^{1/s}( ∫_{0}^{∞}( ∫_{0}^{t} dμ)^{s′−1}( ∫_{0}^{t} dυ)^{1−s′}dμ(t))^{1/s′}, G ↓,

where

s = \frac{γ}{q} > 1, G = ( ∫_{t}^{∞}ψ)^{q/p}, dμ = d(Hf)^{q},

dυ(t) = d(∥χ_{[0,t]}(u/ϕ)∥_{r′r′,ϕ}^{∗})^{γ},

and using Muckenhoupt’s criterion [M], we get

∥Hf∥_{pq,ψ}^{∗} ≪( ∫_{0}^{∞}(∥χ_{[t,∞]}∥_{pq,ψ}^{∗})^{γ}d(∥χ_{[0,t]}(u/ϕ)∥_{r′r′,ϕ}^{∗})^{γ})^{1/γ}

× ( ∫_{0}^{∞}(∥χ_{[0,t]}(u/ϕ)∥_{r′r′,ϕ}^{∗})^{γ(1−s′)}d(Hf)^{qs′})^{1/qs′}

=B_{0}( ∫_{0}^{∞}(Hf)^{r}d(∥χ_{[0,t]}(u/ϕ)∥_{r′r′,ϕ}^{∗})^{1−r})^{1/r}

≪B_{0}( ∫_{0}^{∞}f^{r}ϕ)^{1/r} = B_{0}∥f∥_{rr,ϕ}^{∗}.

Let r > 1 and ∥Hf∥_{pq,ψ}^{∗} ≤ C∥f∥_{rr,ϕ}^{∗} be held for all f ∈ L_{ϕ}^{rr}. Suppose B_{0} < ∞ and let f_{0} be defined by the formula

f_{0}(t) = (∥χ_{[t,∞]}∥_{pq,ψ}^{∗})^{γ/r}(∥χ_{[0,t]}(u/ϕ)∥_{r′r′,ϕ}^{∗})^{γr′/q′r}(u(t)/ϕ(t))^{r′/r},

then ∥f_{0}∥_{rr,ϕ} = B_{0}^{γ/r} < ∞. We may and shall assume for the time being without changing the constant C that supp ψ ⊂ ℝ^{+}. Integrating by parts we find

C∥f_{0}∥_{rr,ϕ}^{∗} = CB_{0}^{γ/r} ≥ ∥Hf_{0}∥_{pq,ψ}^{∗}

= ( ∫_{0}^{∞}(Hf_{0}(t))^{q−1}( ∫_{t}^{∞}ψ)^{q/p}f_{0}(t)u(t) dt)^{1/q}

= (\frac{1}{q} ∫_{0}^{∞}(Hf_{0}(t))^{q}d[ − ( ∫_{t}^{∞}ψ)^{q/p}])^{1/q}.

Employing

(Hf_{0}(t))^{q} = ( ∫_{0}^{t} ( ∫_{0}^{∞}ψ)^{γ/rp}( ∫_{0}^{s} u^{r′}ϕ^{1−r′})^{γ/rq′}u^{r′}(s)ϕ^{1−r′}(s) ds)^{q}

≥ (\frac{q′r}{q(γ + q′r)})^{q}( ∫_{t}^{∞}ψ)^{γq/rp}( ∫_{0}^{t} u^{r′}ϕ^{1−r′})^{γ/r′},

we continue

CB_{0}^{γ/r}

≥ (\frac{q′r}{q(γ + q′r)})(\frac{1}{q} ∫_{0}^{∞}( ∫_{t}^{∞}ψ)^{γq/rp}( ∫_{0}^{t} u^{r′}ϕ^{1−r′})^{γ/r′}d[ − ( ∫_{t}^{∞}ψ)^{q/p}])^{1/q}

= (\frac{q′r}{q(γ + q′r)})(\frac{1}{γ} ∫_{0}^{∞}( ∫_{0}^{t} u^{r′}ϕ^{1−r′})^{γ/r′}d[ − ( ∫_{t}^{∞}ψ)^{γ/p}])^{1/q}

= (\frac{q′r}{q(γ + q′r)})(\frac{1}{r′} ∫_{0}^{∞}( ∫_{t}^{∞}ψ)^{γ/p}d( ∫_{0}^{t} u^{r′}ϕ^{1−r′})^{γ/r′})^{1/q}

= (\frac{q′r}{q(γ + q′r)})(r′)^{−1/q}B_{0}^{γ/q},

and C ≥ (\frac{q′r}{q(γ+q′r)})(r′)^{−1/q}B_{0} follows.

The limiting case r = 1 is proved in [SS]. □

Remark 1.7. (i) Theorem 2.2 has been proved in a different form in [Saw_{1}], Theorem 3 and an alternative proof of it in the case 0 < q < r < ∞, r > 1, p = q, was given in [S_{1}].

(ii) The same theorem for the dual operator follows from the inequalities for nondecreasing functions.

2.2. If p > 0 and ν(t) ≥ 0 is a locally integrable function on ℝ^{+} and the nonincreasing rearrangement on ℝ^{+} of a measurable function f(x) is defined by

f^{∗}(t) = inf {s: meas (x ∈ R^{n}; |f(x)| > s) ≤ t}, t > 0,

then the classical Lorentz spaces Λ_{p}(ν) and Γ_{p}(ν) are defined by

∥f∥_{p,ν}^{∗} = ( ∫_{0}^{∞}(f^{∗}(t))^{p}ν(t) dt)^{1/p} < ∞,

and

∥f∥_{p,ν} = ( ∫_{0}^{∞}(\frac{1}{t} ∫_{0}^{t} f^{∗})^{p}ν(t) dt)^{1/p} < ∞,

respectively. These spaces were introduced by G. G. Lorentz [L_{1}], [L_{2}]. E. Sawyer [Saw_{2}] found that (20) gives a powerful tool for the study of a number of problems in Λ_{p}(ν)-spaces. In particular, it follows from (20) and (21) that Λ_{p}(ν) has the dual space of the form

Λ_{p}^{∗}(ν) = Γ_{p′}((s^{−1}V (s))^{−p′}ν(s)), 1 < p < ∞,

provided V (∞) = ∞ [Saw_{2}], and

Λ_{p}^{∗}(ν) = Γ_{∞}((s^{−p}V (s))^{1/p}), 0 < p ≤ 1.

The same and some other problems for Γ_{p}(ν)-spaces were recently solved in [GHS] by using suitable criteria like (20), (21) for the functions representable as f(x) = \frac{1}{x} ∫_{0}^{x} g, 0 ≤ g ↓. To this end the discretization method from [G_{1}], G_{2}], [G_{3}] was used for the class of functions Ω_{01} of the form

Ω_{01} = \{f(x) ≥ 0, f(x) ↑, \frac{1}{x}f(x) ↓ \},

and a Borel measure dβ on R^{+} with the following nondegeneracy properties

∫_{0}^{∞}(\frac{s}{s + 1})^{p}dβ(x) < ∞; (22)

∫_{0}^{1} dβ(s) = ∫_{1}^{∞}s^{p}dβ(s) = ∞. (23)

Under these conditions the fundamental function of the measure dβ of the form

ρ_{β,p}(t) = ( ∫_{0}^{∞}(\frac{s}{s + 1})^{p}dβ(s))^{1/p}

for any fixed number a > 1 has a discretising sequence {μ_{k}} such that

μ_{0} = 1,
μ_{k+1} = t: min \{\frac{ρ_{β,p}(μ_{k})}{ρ_{β,p}(t)}, \frac{tρ_{β,p}(t)}{μ_{k}ρ_{β,p}(μ_{k})}\} = a, k ≥ 0,
μ_{k−1} = t: min \{\frac{ρ_{β,p}(t)}{ρ_{β,p}(μ_{k})}, \frac{μ_{k}ρ_{β,p}(μ_{k})}{tρ_{β,p}(t)}\} = a, k ≤ 0.

Such a sequence was first used by K. I. Oskolkov and then by a number of authors (see [G_{2}] and the references given there).

Applying the methods developed in [G_{1}], [G_{2}], [G_{3}], we obtain the following Theorem 2.3. Let dβ and dγ be Borel measures such that for dβ the nondegeneracy conditions (22) and (23) are valid. Then for any p > 0 there exists a > 1 and a discretizing sequence {μ_{k}} of the fundamental function ρ_{β,p} such that for 0 < q < p < ∞ the following formula is satisfied

(24)  J ≡ sup_{f∈Ω_{01}} \frac{( ∫_{0}^{∞}f^{q}dγ)^{1/q}}{( ∫_{0}^{∞}f^{p}dβ)^{1/p}} ≈ \{ ∑_{k} [\frac{ρ_{γ,q}(μ_{k})}{ρ_{β,p}(μ_{k})}]^{r}\}^{1/r},

where 1/r = 1/q − 1/p. If 0 < p ≤ q < ∞, then J ≈ sup_{t>0} ρ_{γ,q}(t)/ρ_{β,p}(t).

Now, to obtain analogues of (20) and (21) for functions represented by f(x) = \frac{1}{x} ∫_{0}^{x} g, 0 ≤ g ↓ we consider the particular case of Theorem 2.3, when q = 1 < p < ∞, dβ(s) = s^{−p}ν(s) ds, and {ν_{k}} is the discretizing sequence of the function

(25)  ρ_{β,p}(t) = ( ∫_{0}^{∞} \frac{ν(s) ds}{(s + t)^{p}} )^{1/p} ≡ \frac{1}{t}V^{1/p}(t).

Theorem 2.4. (a) Let 1 < p < ∞, 0 ≤ g(x) ↓, and let the nondegeneracy conditions (23) and (24) be valid for the measure dβ(s) = s^{−p}ν(s) ds, where ν(s) ≥ 0 is a locally integrable function on R^{+}. Then

sup_{0≤f↓} \frac{∫_{0}^{∞}fg}{( ∫_{0}^{∞}(\frac{1}{x} ∫_{0}^{x} f)ν(x) dx)^{1/p}} ≈ ( ∫_{0}^{∞}( ∫_{0}^{x} g)^{p′}V(x) dx)^{1/p′}

where

(26)  V(x) = ∑ δ_{ν_{k}}(x)V^{−p′+1}(x)

and δ_{ν_{k}}(x) is the Dirac δ-function at the point ν_{k}.

(b) Let 0 < p ≤ 1, g(x) ≥ 0. Then

sup_{0≤f↓} \frac{∫_{0}^{∞}fg}{( ∫_{0}^{∞}(\frac{1}{x} ∫_{0}^{x} f)^{p}ν(x) dx)^{1/p}} ≈ sup_{t>0} ( ∫_{0}^{t} g)V^{−1/p}(t).

2.3. This section is devoted to some applications of Theorem 2.4 to the theory of operators in the Γ_{p}(ν)-Lorentz spaces and, in particular, we obtain analogs of some results from [Saw_{2}] proven for Λ_{p}(ν)-spaces. Throughout this section we assume that the nondegeneracy conditions for a weight function ν(s) ≥ 0 of the form

(27)  ∫_{0}^{∞} \frac{ν(s) ds}{(s + 1)^{p}} < ∞; ∫_{0}^{1} s^{−p}ν(s) ds = ∫_{1}^{∞}ν(s) ds = ∞

are satisfied and the measure dV(x) is given by (26).

Theorem 2.5. Let the nondegeneracy conditions (27) be valid for a locally integrable function ν(s) ≥ 0. Then

Γ_{p}^{∗}(ν) = Γ_{p′}(t^{p′}V(t)), 1 < p < ∞,
Γ_{p}^{∗}(ν) = Γ_{∞}(tV^{1/p}(t)), 0 < p ≤ 1.

Now we consider the problem of the boundedness of the Hardy–Littlewood maximal operator

Mf(x) = sup_{x∈Q} \frac{1}{|Q|} ∫_{Q} |f(y)| dy, x ∈ R^{n},

where the supremum is taken over all the cubes Q ⊂ R^{n}, containing the point x and having sides parallel to the coordinate axes. Using the results from [St_{8}], [St_{10}], we obtain the following

Theorem 2.6. Let 1 < p, q < ∞ and let ν(x) ≥ 0, w(x) ≥ 0 be locally integrable functions such that the nondegeneracy conditions (27) are satisfied for ν(x).

(a) If 1 < p ≥ q < ∞, then the inequality

(28)  ∥Mf∥_{q,w} ≤ C∥f∥_{p,ν}, f ∈ Γ_{p,ν}

is valid if, and only if, M = max(A_{0}, A_{1}, B_{0}, B_{1}) < ∞. Moreover, C ≈ M, where

A_{0} = sup_{t>0} ( ∫_{0}^{t} w)^{1/q}( ∫_{t}^{∞}V)^{1/p′},
A_{1} = sup_{t>0} ( ∫_{t}^{∞}s^{−q}w(s) ds)^{1/q}( ∫_{t}^{∞}s^{p′}V(s) ds)^{1/p′},
B_{0} = sup_{t>0} ( ∫_{t}^{∞}s^{−q} log^{q} (\frac{s}{t})w(s) ds)^{1/q}( ∫_{t}^{∞}s^{p′}V(s) ds)^{1/p′},
B_{1} = sup_{t>0} ( ∫_{t}^{∞}s^{−q}w(s) ds)^{1/q}( ∫_{t}^{∞}s^{p′} log^{p′} (\frac{t}{s})V(s) ds)^{1/p′}.

(b) If 1 < q < p < ∞, 1/r = 1/q − 1/p, then the inequality (28) is true if, and only if, L = max(A_{0}, A_{1}, B_{0}, B_{1}) < ∞. Moreover, C ≈ L, where

A_{0} = ( ∫_{0}^{∞}( ∫_{0}^{t} w)^{r/p}( ∫_{t}^{∞}V)^{r/p′}w(t) dt)^{1/r},
A_{1} = ( ∫_{0}^{∞}( ∫_{t}^{∞}s^{−q}w(s) ds)^{r/p}( ∫_{0}^{t} s^{p′}V(s) ds)^{r/p′}t^{−q}w(t) dt)^{1/r},
B_{0} = ( ∫_{0}^{∞}( ∫_{t}^{∞}s^{−q}w(s) ds)^{r/p}( ∫_{0}^{t} s^{p′} log^{p′} (\frac{t}{s})V(s) ds)^{r/p′}t^{−q}w(t) dt)^{1/r},
B_{1} = ( ∫_{0}^{∞}( ∫_{t}^{∞}s^{−q}w(s) ds)^{r/p}( ∫_{0}^{t} s^{p′} log^{p′} (\frac{t}{s})V(s) ds)^{r/q′}t^{p′}V(t) dt)^{1/r}.

(c) If 0 < p ≤ 1 < q < ∞, then the inequality (28) is valid if, and only if, N = max(G_{0}, G_{1}.G_{2}) < ∞. Moreover, C ≈ N, where

G_{0} = sup_{t>0} ( ∫_{0}^{t} w)^{1/q}V^{−1/p}(t),

G_{1} = sup_{t>0} ( ∫_{t}^{∞}s^{−q}w(s) ds)^{1/q}tV^{−1/p}(t),

G_{2} = sup_{t>0} ( ∫_{t}^{∞}s^{−q} log^{q} (\frac{s}{t})w(s) ds)^{1/q}tV^{−1/p}(t).

An analogous assertion holds for the Hilbert transform

Hf(x) = p.v. ∫_{−∞}^{∞} \frac{f(y) dy}{x − y} .

Theorem 2.7. Let the hypotheses and the notation of Theorem 2.6 be preserved.

(a) If 1 < p ≤ q < ∞, then the inequality

(29)  ∥Hf∥_{q,w} ≤ C∥f∥_{p,ν}, f ∈ Γ_{p,ν}

is valid if, and only if,

M_{H} = max(A_{0}, A_{1}, B_{0}, B_{1}, D_{0}, D_{1}) < ∞.

Moreover, C ≈ M_{H}, where

D_{0} = sup_{t>0} ( ∫_{0}^{t} w)^{1/q}( ∫_{t}^{∞}log^{p′} (\frac{s}{t})V(s) ds)^{1/p′},

D_{1} = sup_{t>0} ( ∫_{0}^{t} log^{q} (\frac{t}{s})w(s) ds)^{1/q}( ∫_{t}^{∞}V)^{1/p′}.

(b) If 1 < q < p < ∞, 1/r = 1/q − 1/p, then (29) is true if, and only if, L_{H} = max(A_{0}, A_{1}, B_{0}, B_{1}, D_{0}, D_{1}) < ∞. Moreover, C ≈ L_{H}, where

D_{0} = ( ∫_{0}^{∞}( ∫_{0}^{t} w)^{r/p}( ∫_{t}^{∞}log^{p′} (\frac{s}{t})V(s) ds)^{r/p′}w(t) dt)^{1/r},

D_{1} = ( ∫_{0}^{∞}( ∫_{0}^{t} log_{q} (\frac{t}{s})w(s) ds)^{r/q}( ∫_{t}^{∞}V)^{r/q′}V(s) dt)^{1/r}.

(c) If 0 < p ≤ 1 < q < ∞, then (29) is fulfilled if, and only if, N_{H} = max(G_{0}, G_{1}, G_{3}, G_{4}) < ∞. Moreover, C ≈ N_{H}, where

G_{3} = sup_{t>0} ( ∫_{0}^{t} log^{q} (\frac{s}{t})w(s) ds)^{1/q}V^{−1/p}(t).

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