EXTRAPOLATION OF REDUCED

SOBOLEV IMBEDDINGS

Miroslav Krbec and Hans-Jürgen Schmeisser

1. Background

The limiting imbeddings of Sobolev spaces attract a lot of attention nowadays. Various settings of the problem have been recently studied in many papers (cf. [FLS], [ET1], [EK], [KSchm], [KScho], …). The celebrated limiting imbedding theorem [Tru], [Po], [St] states in its basic form that the Sobolev space W_{p}^{m}(Ω), where Ω ⊂ ℝ^{N} is a bounded domain with a sufficiently smooth boundary, 1 < p < ∞, and mp = N, is imbedded into the Orlicz space L_{Φ}(Ω) with Φ(t) = exp t^{N/(N−m)} − 1. The key to the proof is to expand Φ into its Taylor’s series, taking into account the definition of the Luxemburg norm and showing that

(1.1)  ∥u | L_{q}∥ ≤ cq^{1−m/N}∥u | W_{p}^{m}∥, q large.

Equivalently, one can write

(1.1’)  sup_{1<q<∞} q^{−(1−m/N)}∥u | L_{q}∥ ≤ c∥u | W_{p}^{m}∥, q large,

and it is easy to check that the expression on the left hand side of (1.1’) is an equivalent norm in L_{Φ}(Ω) (cf. [ET2]). Similarly, the condition

lim_{p→∞} \frac{1}{p}(∫_{Ω} |u(x)|^{Np/(N−m)} dx)^{1/p} < ∞,

can be used as in the recent paper [FLS].

The conditions (1.1) and (1.1’) express a control of the blow up of ∥u | L_{q}∥ when q tends to ∞ (as is well known, W_{p}^{m}(Ω) ↛ L_{∞}(Ω) when mp = N). From the point of view of the scale of Lebesgue spaces, it is of course possible to transfer the situation to the case q ↗ q_{0} < ∞ and to balance the speed of blow up of ∥u | L_{q}∥. Provided that the speed at which ∥u | L_{q}∥ tends to infinity can be controlled in a suitable way by a power function, one gets the logarithmic Lebesgue spaces L_{q_{0}} log^{σ}L for negative σ’s. The details can be found in [ET2], and from the point of view of this approach it is quite natural to use the term logarithmic L_{∞} spaces for the exponential Orlicz spaces. More tedious considerations mimicking essentially the arguments from [ET2] make it possible to obtain a “finer logarithmic tuning” of the scale near L_{q_{0}} (q_{0} < ∞) and near L_{∞}. Let us still observe that the above mentioned results can be expressed also in the language of suitable extrapolation methods (see [M] for the definitions of the Δ and Σ methods). The basic limiting situation repeats in a changed setting when considering W_{p}^{1+m}, mp = N. A function from W_{p}^{1+m} is generally not Lipschitz continuous, even though it is Hölder continuous with all 0 < α < 1. It has been shown by Brézis and Wainger [BW] that a better target space can be found, namely, the space Lip / log^{1/p′}, and this was generalized to Bessel potential logarithmic Sobolev spaces (with a non-integer smoothness parameter) in [EK]; at the same time a natural link between these two kinds of limiting imbeddings was found here.

There are far less results on the weighted limiting imbeddings; they are the subject of a recent paper [KScho].

In this contribution we shall follow another line of limiting considerations, which has been discussed in [KSchm]. The point here is a weakening of the assumptions on the smoothness: It is known that the Sobolev imbedding theorem holds with unchanged exponents if some of the highest order derivatives are neglected. Let M(N, m) be a subset of all N-tuples α = (α_{1}, …, α_{N}), defined by

M(N, m) = {α: |α| = m, α_{i} = 0 or α_{i} = 1 for 1 ≤ i ≤ N}

and let M ⊂ M(N, m). We define the reduced Sobolev space W_{p}^{M}(ℝ^{N}), 1 < p < ∞, as

W_{p}^{M} = W_{p}^{M}(ℝ^{N}) = {f ∈ L_{p}: D^{β}f ∈ L_{p} for all β ≤ α and all α ∈ M}.

In the paper [Ad] there are considered sublimiting imbeddings for these spaces with M = M(N, m) and with subsets M, satisfying the conditions

∑_{α∈M} α_{i} = k ≥ 1 for all i = 1, …, N.

It has been proved in [Ad] and [BIN] that then the sublimiting imbeddings hold with the same target space as in the case of W_{p}^{m}, that is,

W_{p}^{M(N,m)}(ℝ^{N}) ↪ L_{q}(ℝ^{N}) for q = Np/(N − mp).

Related estimates for more general subsets M and mixed norm spaces can be found in [BIN, Theorem 15.7].

Here we are concerned with more general spaces W M ; \overline{p}(ℝ^{N}) corresponding to mixed norms and

M = S(N, (n_{k})) = S(N, n_{1}, …, n_{m})

= {α = (α^{1}, …, α^{m}): α^{j} = (α_{1}^{j}, …, α_{n_{j}}^{j}), |α^{j}| = 1, 1 ≤ j ≤ m},

where n_{1} + ⋯ + n_{m} = N (see Definition 3.2 and Remark 3.3).

For instance, in the non-mixed case p = N/m = n_{1} = ⋯ = n_{m} we prove that

W_{p}^{M}(ℝ^{N}) ↪ L_{Φ}(ℝ^{N}) with Φ(t) ∼ exp t^{N/m(N−m)},

see Corollary 3.8 (ii).

The paper is divided into two major sections. In Section 2 we use an alternative classical analysis approach for some particular cases to get estimates of ∥u | L_{q}∥ when q → ∞. We get a sharp result, however, there are natural limits as to the generality in order that the proofs do not become technically unduly difficult and lengthy and, also, only particular subsets of integer derivatives up to a given order come into the play. The core of this contribution is Section 3, using the Fourier analysis approach to the fractional Sobolev spaces with dominating mixed derivatives as developed in [Sc1], [ScT, Chapter 2]. In particular, Theorem 3.5 is our most general result, covering especially the classical setting of the problem. Due to the limited extent of this paper we shall restrict ourselves only to the spaces of Lizorkin-Triebel type (the F -spaces); we refer to [KSchm] for the B-spaces. The crucial point is that

W M ; \overline{p}(ℝ^{N}) = S_{\overline{p},\overline{2}}^{(1,...,1)}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})

(see Definition 3.1) if M = S(N, n_{1}, …, n_{m}), N = n_{1} + ⋯ + n_{m}, which enables us to prove imbeddings for the reduced Sobolev spaces via spaces with dominating mixed derivatives.

The spaces we consider are rather general, however, it should be pointed out that this is not so for the sake of the generality itself. Let us look more closely at the following illustrative example (see Definition 3.1 for the spaces in question). Put N = 3, m = 2, \overline{p} = (3/2, 3/2, 3/2) and consider the space S_{\overline{p},\overline{2}}^{(1,1)}F (ℝ^{1} ×ℝ^{2}). Clearly, if f_{1} ∈ W_{3/2}^{1}(ℝ^{1}), f_{2} ∈ W_{3/2}^{1}(ℝ^{2}) and f = f_{1} ⊗f_{2}, then, in general, f ∉ L_{q} for large q’s. On the other hand, if \overline{p} = (1, 2, 2), then S_{\overline{p},\overline{2}}^{(1,1)}F (ℝ^{1} × ℝ^{2}) ↪ L_{Φ} with Φ(t) ∼ exp t^{2}. We observe that

S_{\overline{p},\overline{2}}^{(1,1)}F = \{f: D^{(1,1,0)}f, D^{(1,0,1)}f, D^{(1,0,0)}f, D^{(0,1,0)}f, D^{(0,0,1)}f, f ∈ H\overline{p}\},

where H\overline{p} = S_{\overline{p},\overline{2}}^{(0,0)}F is an appropriate substitute for L\overline{p}. This shows that the seemingly too general mixed norm setting is in fact necessary and quite natural.

2. The classical technique approach

Here we shall deal with the spaces W_{p}^{S}(ℝ^{N}) with N = 2k, k ≥ 2, p = N/2, and S = S(k) ⊂ M(N, 2) satisfying

(2.1)  ∑_{α∈S} α_{j} = k, j = 1, …, N.

The condition (2.1) means that if we write the elements of S as rows of a matrix, then the sum in each of the columns equals k.

A generalization of the considerations in this section to higher order spaces is possible, the proofs and the notation would become more complicated, the underlying idea being the same.

For j = 1, …, N, let S_{j} = {l ≠ j: α_{l} = α_{j} = 1 for some α ∈ S}.

For an N-tuple \overline{p} = (p_{1}, …, p_{N}) with 0 < p_{i} ≤ ∞ we shall denote by ∥u | L\overline{p}(ℝ^{N})∥ the norm in the Lebesgue mixed norm space L\overline{p}(ℝ^{N}).

For 1 < p ≤ ∞ the symbol ∥u | L_{p}(S_{j})∥ will stand for the L_{p} norm of u with respect to the variables (x_{l})_{l∈S_{j}}. If S′_{j} is the complement of S_{j} with respect to {1, …, N}, then the meaning of ∥u | L_{q_{1}}(S_{j}) | L_{q_{2}}(S′_{j})∥ is that we take first the L_{q_{1}} norm of u with respect to the variables in S_{j} and then the L_{q_{2}} norm of u with respect to the remaining variables.

We observe that by Minkowski’s inequality, the value of ∥u | L\overline{p}(ℝ^{N})∥ does not decrease after interchanging the order of integration with respect to x_{i} and x_{i+1} provided that p_{i} ≤ p_{i+1}.

The symbol L_{Φ} = L_{Φ}(ℝ^{N}) will stand for the (classical) Orlicz space in the sense of [KR], and E_{Φ} = E_{Φ}(ℝ^{N}) will denote the closure of C_{0}^{∞} in L_{Φ}. It is well known that L_{Φ} = E_{Φ} if and only if the function Φ satisfies the Δ_{2}-condition, which is of course not the case of the exponential Orlicz spaces.

Now we are ready to state and to prove the following reduced limiting theorem.

Theorem 2.1. Let k ≥ 2 be an integer, N = 2k, and let S be a subset of M satisfying the condition (2.1). If u ∈ W_{k}^{S} is supported in a unit ball, then u ∈ E_{Φ}(ℝ^{N}) with Φ(t) = exp t^{N/2(N−1)} − 1 and there exists c > 0 independent of u such that

∥u | E_{Φ}∥ ≤ c ∑_{α∈S} ∥D^{α}u | L_{N/2}∥.

Proof. Since there is no anisotropy as far as the integrability properties are concerned, the order of variables is not relevant. Hence after a suitable renumbering of variables we can switch to the following basic situation: For l = 1, …, N/2 let S_{l} be the N/2-tuple whose j-th component equals j + N/2, 1 ≤ j ≤ N/2. The N/2-tuples S_{l} for l > N/2 become all equal to (1, …, N/2) and it is obvious that among the couples (l, j + N/2), 1 ≤ l ≤ N/2, 1 ≤ j ≤ N/2, there are already all couples of indices with respect to which the second order derivatives of u ∈ W_{N/2}^{S} are supposed to exist and to belong to L_{N/2}; altogether there are N^{2}/4 different couples when counted without respect to the order of their components.

It is also enough to assume that u ∈ C_{0}^{∞}. First consider u as a function in variables x_{1}, …, x_{1+N/2} and invoke the classical limiting imbedding [Tru]; we have

(2.2)  ∥u | L_{r}(S_{1+N/2})∥ ≤ cr^{1−2/N} ∑_{j=1}^{N/2} ∥D_{j}u | L_{N/2}(S_{j+N/2})∥.

Taking the L_{r} norms with respect to the remaining variables, we obtain from (2.2) that

(2.3)  ∥u | L_{r}(ℝ^{N})∥ ≤ cr^{1−2/N} ∑_{j=1}^{N/2} ∥D_{j}u | L_{N/2}(S_{j+N/2}) | L_{r}(S_{1})∥.

Now the same argument applies to each derivative D_{j}u from the right hand side of (2.3) since D_{j}u (1 ≤ j ≤ N/2) as functions in x_{1+N/2}, …, x_{N} have all first order derivatives. Hence

∥D_{j}u | L_{r}(S_{1})∥ ≤ cr^{1−2/N} ∑_{j=1+N/2}^{N} ∥D_{i}D_{j}u | L_{N/2}(S_{1})∥

and

(2.4)  ∥D_{j}u | L_{r}(S_{1}) | L_{N/2}(S_{1+N/2})∥

≤ cr^{1−2/N} ∑_{j=1+N/2}^{N} ∥D_{i}D_{j}u | L_{N/2}(ℝ^{N})∥.

The repeated use of Minkowski’s inequality gives the natural order of variables on the left hand side of (2.4). Combining this together with (2.3) we get

(2.5)  ∥u | L_{r}(ℝ^{N})∥ ≤ cr^{2(1−2/N)}∥u | W_{N/2}^{S}(ℝ^{N})∥.

The proof can be now concluded in a standard way, namely by expanding the function Φ(t) = exp t^{N/2(N−1)} − 1 into Taylor’s series. □

Remark 2.2. Another natural idea to prove Theorem 2.1 seems to be a combinatorial refinement of Gagliardo’s and Adams’ technique from [Ad]. Nevertheless, a thorough analysis reveals that even though a certain exponential inequality can be proved, the target space for the resulting imbedding is worse. To be more specific, for spaces of second order such approach consists of going through estimates which give

∥u | L_{q}∥ ≤ cq ∑_{i=1}^{N} ∥D_{i}u | L_{N/2}(S_{j+N/2}) | L_{q}(S_{1})∥

and

∥D_{i}u | L_{q}(S_{1}) | L_{N/2}(S_{1+N/2})∥ ≤ cq ∑_{j∈S_{i}} ∥D_{j}D_{i}u | L_{N/2}(ℝ^{N})∥,

which pasted together yield only

∥u | L_{q}(ℝ^{N})∥ ≤ cq^{2}∥u | W_{N/2}^{S}(ℝ^{N})∥,

that is, an estimate worse for all dimensions N than that in (2.5).

3. The Fourier analysis approach

Let N ≥ 2 and ℝ^{N} = ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}. A point x ∈ ℝ^{N} will be written as x = (x^{1}, …, x^{m}) with x^{j} ∈ ℝ^{n_{j}}, 1 ≤ j ≤ m. Let \overline{p} = (p_{1}, …, p_{m}), 0 < p_{i} ≤ ∞. We put

∥f | L\overline{p}(ℝ^{N})∥

= ( ∫_{ℝ^{nm}} ( … ( ∫_{ℝ^{n_{1}}} |f(x^{1}, …, x^{m})|^{p_{1}} dx^{1})^{p_{2}/p_{1}} …)^{p_{m}/p_{m−1}} dx^{m})^{1/p_{m}}

and (κ = (κ_{1}, …, κ_{m}))

∥a_{κ} | ℓ\overline{p}∥ = ( ∑_{κ_{m}=0}^{∞}( … ( ∑_{κ_{1}=0}^{∞} |a_{κ_{1},...,κ_{m}}|^{p_{1}})^{p_{2}/p_{1}} …)^{p_{m}/p_{m−1}})^{1/p_{m}}

(with the usual modification if p_{j} = ∞).

Let {ϕ_{k}^{j}(ξ^{j})} be the standard resolution of unity associated to dyadic balls in ℝ^{n_{j}}, 1 ≤ j ≤ m, that is, a sequence of functions {ϕ_{k}^{j}} such that

∑_{k=0}^{∞} ϕ_{k}^{j}(ξ^{j}) ≡ 1 in ℝ^{n_{j}},

supp ϕ_{0}^{j} ⊂ {ξ^{j} ∈ ℝ^{n_{j}}: |ξ^{j}| ≤ 2},

supp ϕ_{k}^{j} ⊂ {ξ^{j} ∈ ℝ^{n_{j}}: 2^{k−1} ≤ |ξ^{j}| ≤ 2^{k+1}},

with ϕ_{k}^{j}(ξ^{j}) = ϕ^{j}(2^{−k}ξ^{j}), k = 1, 2, …, where ϕ is a C^{∞} function whose support is contained in {1/2 ≤ ξ^{j} ≤ 2}.

Let S′(ℝ^{N}) be the space of tempered distributions, and let F and F^{−1} denote the Fourier transform and its inverse, respectively. For f ∈ S′(ℝ^{N}) we put

f_{k}(x) = f_{k_{1},...,k_{m}}(x^{1}, …, x^{m}) := (F^{−1}(ϕ_{k_{1}}^{1} ⊗ ⋯ ⊗ ϕ_{k_{m}}^{m})Ff)(x^{1}, …, x^{m}),

where k = (k_{1}, …, k_{m}), k_{j} = 0, 1, 2, …, j = 1, …, m, x ∈ ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}.

Definition 3.1. Let \overline{p} = (p_{1}, …, p_{m}), \overline{q} = (q_{1}, …, q_{m}), \overline{r} = (r_{1}, …, r_{m}) with 0 < p_{j} < ∞, 0 < q_{j} ≤ ∞, and −∞ < r_{j} < ∞, j = 1, …, m. Then we define the Lizorkin-Triebel space with dominating mixed smoothness

S\overline{r} ; \overline{p},\overline{q}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}) = \{f ∈ S′(ℝ^{N}):

∥f | S\overline{r} ; \overline{p},\overline{q}F ∥ = ∥2^{k_{1}r_{1}+···+k_{m}r_{m}}f_{k_{1},...,k_{m}}(x) | ℓ\overline{p} | L\overline{q}∥ < ∞\}.

We shall use the notation \overline{k} = (k, …, k) (an m-tuple).

Definition 3.2. Let \overline{p} = (p_{1}, …, p_{m}), 1 < p_{j} < ∞, and −∞ < r_{j} < ∞, j = 1, …, m. The fractional Sobolev space with dominating mixed derivatives is defined as

S\overline{r} ; \overline{p}H(ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}) = \{f ∈ S′(ℝ^{N}):

∥f | S\overline{r} ; \overline{p}H∥ = ∥F^{−1}(1 + |ξ^{1}|^{2})^{r_{1}/2} … (1 + |ξ^{m}|^{2})^{r_{m}/2}Ff | L\overline{p}∥ < ∞\}.

Remark 3.3. The F -spaces of the above type have been intensively studied in [Sc1]. The model case N = 2, n_{1} = n_{2} = 1 can be found in [ScT, Chapter 2]. All the spaces are quasi-Banach spaces (Banach spaces if min(p_{j}, q_{j}) ≥ 1) and independent of the choice of the systems {ϕ^{j}} (up to equivalence of the resulting quasi-norms). If 0 < p_{1} = ⋯ = p_{m} < ∞,

0 < q_{1} = ⋯ = q_{m} ≤ ∞, n_{1} = ⋯ = n_{m} = 1, then S\overline{r} ; \overline{p},\overline{q}F (ℝ^{N}) is a special

case of non-homogeneous general function spaces in the sense of [Tri3–5].

The corresponding spaces of Besov type can be considered, too. Limiting imbeddings for them have been proved in [KSchm]. For the spaces S\overline{r} ; \overline{p},\overline{q}B(ℝ^{N}) with n_{1} = ⋯ = n_{m} = 1, r_{j} ≥ 0, p_{j} ≥ 1, q_{j} ≥ 1 we refer to the book by Amanov [Am] and the literature quoted there. The remark on non-homogeneous general function spaces concerning [Tri3–5] applies as well.

Let r_{j} be non-negative integers and 1 < p_{j} < ∞, j = 1, …, m. We define the Sobolev space with dominating mixed derivatives

S\overline{r} ; \overline{p}W (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}) = \{f ∈ L\overline{p}(ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}):

∥f | S\overline{r} ; \overline{p}W ∥ = ∑_{\begin{matrix}θ_{j}=0,1; |α^{j}|=r_{j} \\ j=1,...,m \end{matrix}} ∥D^{(θ_{1}α^{1},...,θ_{m}α^{m})}f | L\overline{p}∥ < ∞\}.

Using the arguments from [Sc1, Thm. A.2] (see also [Sc2, Thm. 3] and [ScT, Thm. 2.3.1] for the case n_{1} = ⋯ = n_{m}) one can prove that S\overline{r} ; \overline{p},\overline{2}F (ℝ^{N}) = S\overline{r} ; \overline{p}H(ℝ^{N}) for 1 < p_{j} < ∞ and −∞ < r_{j} < ∞ (1 ≤ j ≤ m) with equivalent norms. Moreover, if r_{j} are non-negative integers, then

S\overline{r} ; \overline{p},\overline{2}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}) = S\overline{r} ; \overline{p}W (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}).

We observe that by standard arguments (Fourier multipliers of Mikhlin-Hörmander type) the norm in S\overline{r} ; \overline{p}W (ℝ^{N}) is equivalent to

∑_{\begin{matrix}0≤|α^{j}|≤r_{j} \\ j=1,...,m \end{matrix}} ∥D^{(α^{1},...,α^{m})}f | L\overline{p}∥.

The space S\overline{1} ; \overline{p}W (ℝ^{N}) is of special interest: It consists of those f ∈ L\overline{p}(ℝ^{N}) for which D^{(α^{1},...,α^{m})}f ∈ L\overline{p}(ℝ^{N}) and |α^{j}| = 0 or |α^{j}| = 1 for all j = 1, …, m. Its special non-mixed case is the reduced Sobolev space. This also establishes connections with the spaces W_{p}^{M}(ℝ^{N}) with M = S(N, n_{1}, …, n_{m}) introduced earlier.

In what follows we shall pursue a finer analysis of the asymptotic behaviour of the constant c = c(q) in the imbedding inequality

∥f | L\overline{q}(ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥ ≤ c ∥f | S\overline{r} ; \overline{p},\overline{2}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥

when r_{j} = n_{j}/p_{j} (j = 1, …, m) and q → ∞. Let us recall that, in general,

(3.1)  S\overline{r} ; \overline{p},\overline{q}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}) ⊂ S\overline{t} ; \overline{u},\overline{v}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})

provided that 0 < p_{j} < u_{j} < ∞, 0 < q_{j}, v_{j} ≤ ∞, and

r_{j} − \frac{n_{j}}{p_{j}} = t_{j} − \frac{n_{j}}{u_{j}} (j = 1, …, m).

In particular, we have

(3.2)  S\overline{r} ; \overline{p},\overline{q}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}) ⊂ L\overline{u}(ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})

if r_{j} ≥ n_{j}(1/p_{j} − 1/u_{j}) and 1 < p_{j} < u_{j} < ∞ (j = 1, …, m).

Proofs of the imbeddings (3.1) and (3.2) can be found in [ScT, Thm. 2.4.1] (the model case n_{1} = n_{2} = 1, m = 2); a detailed proof in the general case can be found in [Sc1, Chapter 6] or in [Sc3]. These imbeddings are sharp and cannot be improved with respect to the scale of the spaces involved. This follows from [SiT], taking into account that

∥f | S\overline{r} ; \overline{p},\overline{q}F ∥ ∼ ∏_{j=1}^{m} ∥f^{j} | F_{p_{j},q_{j}}^{r_{j}}(ℝ^{n_{j}})∥

for all f = f^{1} ⊗ ⋯ ⊗ f^{m}, where f^{j} ∈ F_{p_{j},q_{j}}^{r_{j}}(ℝ^{n_{j}}).

Of crucial importance for establishing the limiting imbeddings for the F -spaces are Lemma 3.4 and Theorem 3.5. We shall not give the full proofs; an interested reader can look into [KSchm].

Lemma 3.4. Let 1 < p_{j} < q_{j} < ∞, r_{j} = n_{j}/p_{j} (j = 1, …, m). If 2 ≤ p_{1} ≤ p_{2} ≤ ⋯ ≤ p_{m}, then there exists a constant c independent of \overline{q} and f such that

∥f | L\overline{q}(ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥ ≤ c ∏_{j=1}^{m} q_{j}^{1−1/p_{j}}∥f | S\overline{r} ; \overline{p},\overline{2}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥

for all f ∈ S\overline{r} ; \overline{p},\overline{2}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}).

Proof. Step 1. Let f_{k}(x) = f_{k_{1},...,k_{m}}(x^{1}, …, x^{m}) be the function from Definition 3.1. Then

(3.3) ∥f | L\overline{q}∥ = ∥ ∑_{k_{m}=0}^{∞} … ∑_{k_{1}=0}^{∞} f_{k_{1},...,k_{m}}(x^{1}, …, x^{m}) | L\overline{q}∥
≤ ∥f_{k_{1},...,k_{m}}(x^{1}, …, x^{m}) | ℓ_{1|k_{1}}(L_{q_{1}}) | ⋯ | ℓ_{1|k_{m}}(L_{q_{m}})∥,

where ℓ_{κ_{j}|k_{j}}(L_{q_{j}}) denotes the L_{q_{j}} norm with respect to x^{j} ∈ ℝ^{n_{j}} and the ℓ_{κ_{j}} norm with respect to k_{j} (the subindex k_{j} will be sometimes omitted if a misunderstanding cannot occur). Step 2. An inequality of Nikol’skii type (see [ScT, Proposition 2.4.1 (i)]) applied to the right hand side of (3.3) gives ∥f | L\overline{q}∥

≤ c∥∏_{j=1}^{m} 2^{k_{j}n_{j}(1/p_{j}−1/q_{j})}f_{k_{1},...,k_{m}}(x^{1}, …, x^{m}) | ℓ_{1|k_{1}}(L_{p_{1}}) | ⋯ | ℓ_{1|k_{m}}(L_{p_{m}})∥

with c independent of \overline{q}. Using Hölder’s inequality m-times we arrive at

∥f | L\overline{q}∥ ≤ c ∏_{j=1}^{m}( ∑_{k_{j}=0}^{∞} 2^{−k_{j}n_{j}p′_{j}/q_{j}})^{1/p′_{j}}×

× ∥∏_{j=1}^{m} 2^{k_{j}n_{j}/p_{j}}f_{k_{1},...,k_{m}} | ℓ_{p_{1}}(L_{p_{1}}) | ⋯ | ℓ_{p_{m}}(L_{p_{m}})∥,

where 1/p_{j} + 1/p′_{j} = 1 and c is independent of f and \overline{q}.

Step 3. Since ∑_{ℓ=0}^{∞} (2^{−np′})^{ℓ/q} ∼ q for q → ∞ (with n and p′ fixed), we get

(3.4) ∥f | L\overline{q}∥ ≤ c ∏_{j=1}^{m} q_{j}^{1−1/p_{j}}×

× ∥∏_{j=1}^{m} 2^{k_{j}n_{j}/p_{j}}f_{k_{1},...,k_{m}} | ℓ_{p_{1}}(L_{p_{1}}) | ⋯ | ℓ_{p_{m}}(L_{p_{m}})∥.

If p_{j} ≤ p_{k}, then the ℓ_{p_{k}}(L_{p_{j}})-norm is majorized by the L_{p_{j}}(ℓ_{p_{k}})-norm and the same relation holds between the ℓ_{p} and ℓ_{2} norms provided that p ≥ 2. Therefore, it follows from (3.4) that

∥f | L\overline{q}∥ ≤ c ∏_{j=1}^{m} q_{j}^{1−1/p_{j}}∥f | S\overline{r} ; \overline{p},\overline{2}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥.

□

Theorem 3.5. Let 1 ≤ p_{j} < q_{j} < ∞, r_{j} = n_{j}/p_{j} (j = 1, …, m). Then there exists a constant c independent of \overline{q} and f such that

(3.5)  ∥f | L\overline{q}(ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥

≤ c ∏_{j=1}^{m} q_{j}^{1−1/p_{j}}∥f | S\overline{r} ; \overline{p},\overline{2}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥

for all f ∈ S\overline{r} ; \overline{p},\overline{2}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}).

Sketch of the proof. It is enough to consider only m = 2. The proof goes along the same lines if m > 2, but it is more complicated from the technical point of view. Moreover, a complete proof even in this simplified form would be rather long because it uses a lot of non-folklore arguments from the interpolation theory. Therefore we restrict ourselves to explaining the main points of the proof. Step 1. Four special cases can be distinguished: (1) p_{1} = p_{2} = 1, (2) p_{1} = 1, p_{2} ≥ 2, (3) p_{1} ≥ 2, p_{2} = 1, and (4) 2 ≤ p_{1} ≤ p_{2}. The remaining combinations of p_{1} and p_{2} can be handled by complex interpolation. Indeed,

(3.6)  [S\overline{r} ; \overline{p},\overline{q}F, S\overline{t} ; \overline{u},\overline{v}F]_{θ} = S\overline{c} ; \overline{a},\overline{b}F

for 1 ≤ p_{j}, q_{j}, u_{j}, v_{j} < ∞, −∞ < r_{j}, t_{j} < ∞ (j = 1, …, m), 0 < θ < 1, and

\frac{1}{a_{j}} = \frac{1 − θ}{p_{j}} + \frac{θ}{u_{j}}, \frac{1}{b_{j}} = \frac{1 − θ}{q_{j}} + \frac{θ}{v_{j}}, c_{j} = (1 − θ)r_{j} + θt_{j} (j = 1, …, m). The

proof of (3.6) goes via iteration and retraction to the complex interpolation of vector-valued L_{p}-spaces and vector-valued sequence spaces, which is dealt with in [Tri1, Thms. 1.18.1, 1.18.4, and 2.4.1 (d)].

The case (4) was considered in Lemma 3.4. The core of the proof for the remaining three cases consists in showing that

(3.7)  ∥f | L\overline{∞}∥ ≤ c∥f | S_{\overline{1},\overline{2}}^{(n_{1},n_{2})}F ∥,

(3.8)  ∥f | L\overline{1}∥ ≤ c∥f | S_{\overline{1},\overline{2}}^{(n_{1},n_{2})}F ∥,
(3.9)  ∥f | L_{(1,∞)}∥ ≤ c∥f | S_{\overline{1},\overline{2}}^{(n_{1},n_{2})}F ∥,
(3.10)  ∥f | L_{(∞,1)}∥ ≤ c∥f | S_{\overline{1},\overline{2}}^{(n_{1},n_{2})}F ∥.

Step 2. We shall give a sketch of the proof of (3.7). Combining an idea from [Ja] with our technique to prove Lemma 3.4, we get, applying the inequality of Nikol’skii type ([ScT, Prop. 2.4.1 (i)], with \overline{q} = \overline{∞} and \overline{p} = \overline{2}), that

(3.11)  ∥f | L\overline{q}∥ ≤ ∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}(L_{∞}) | ℓ_{2}(L_{∞})∥
≤ ∥2^{k_{1}n_{1}/2}2^{k_{2}n_{2}/2}f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}(L_{2}) | ℓ_{1}(L_{2})∥
= ∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}^{n_{1}/2}(L_{2}) | ℓ_{1}^{n_{2}/2}(L_{2})∥.

If 0 < u < 1 < v < 2 and 0 < θ < 1 are such that

\frac{1 − θ}{u} + \frac{θ}{v} = 1

and if we define r and t by

n_{2} − \frac{n_{2}}{u} = r − \frac{n_{2}}{2} and n_{2} − \frac{n_{2}}{v} = t − \frac{n_{2}}{2} ,

then

(1 − θ)r + θt = \frac{n_{2}}{2} .

By real interpolation (see [Pe, Chapter 5, Thms. 2,4] and for special cases also [Tri1, Sections 1.18.2, 1.18.4, 1.18.6] and [BL, Sections 5.2 and 5.6]), we get that

(3.12)  (ℓ_{∞}^{r}(L_{2}(A)), ℓ_{∞}^{t}(L_{2}(A)))_{θ,1} = ℓ_{1}^{n_{2}/2}(L_{2}(A)),

where A = ℓ_{1}^{n_{1}/2}(L_{2}).

Step 1 of the proof of [ScT, Prop. 2.4.1] shows that (for fixed k_{2})

∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}^{n_{1}/2}(L_{2}) | L_{2}∥
≤ c∥2^{k_{2}n_{2}(1/u−1/2)}f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}^{n_{1}/2}(L_{2}) | L_{u}∥.

Multiplying by 2^{k_{2}r} and taking sup-norms with respect to k_{2} at first on the right hand side and then on the left hand side, we get

(3.13) ∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}^{n_{1}/2}(L_{2}) | ℓ_{∞}^{r}(L_{2})∥
≤ c∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}^{n_{1}/2}(L_{2}) | L_{u}(ℓ_{∞}^{n_{2}})∥,

and, moreover, the same inequality holds with v and t instead of u and r, respectively. One more invoking of real interpolation shows that

(3.14)  (L_{u}(B), L_{v}(B))_{θ,1} = L_{1}(B),

where B = ℓ_{∞}^{n_{2}}(ℓ_{1}^{n_{1}/2}(L_{2})). The interpolation formulas (3.12) and (3.14) can be applied to systems {f_{k^{1},k_{2}}(x^{1}, x^{2})} of entire analytic functions (cf. [Tri 2, Subsection 2.4.9] or [Pä] for the complex method). Consequently, we get from (3.13) that

(3.15) ∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}^{n_{1}/2}(L_{2}) | ℓ_{1}^{n_{2}/2}(L_{2})∥
≤ c∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}^{n_{1}/2}(L_{2}) | L_{1}(ℓ_{∞}^{n_{2}})∥.

If u, v, θ are as above and r and t are given by

n_{1} − \frac{n_{1}}{u} = r − \frac{n_{1}}{2} and n_{1} − \frac{n_{1}}{v} = t − \frac{n_{1}}{2} ,

respectively, then (1 − θ)r + θt = n_{1}/2.

For fixed x^{2}, k_{1}, k_{2} we shall use Nikol’skii’s inequality with respect to x^{1} (cf. [ScT, Thm. 1.6.2]) to show that

(3.16)  ∥2^{k_{2}n_{2}}f_{k_{1},k_{2}}(x^{1}, x^{2}) | L_{2|x^{1}}∥
≤ c∥2^{k_{1}n_{1}(1/u−1/2)}2^{k_{2}n_{2}}f_{k_{1},k_{2}}(x^{1}, x^{2}) | L_{u|x^{1}}∥,

where L_{u|x^{1}} indicates that the L_{u}-(quasi-)norm is to be taken with respect to x^{1}. Multiplying (3.16) by 2^{k_{1}r} and taking sup-norms with respect to k_{1}, k_{2}, we get

(3.17)  ∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | L_{2|x^{1}} | ℓ_{∞|k_{2}}^{n_{2}} | ℓ_{∞|k_{1}}^{r}∥

≤ c∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ(n_{1},n_{2}) ; \overline{∞} | L_{u|x^{1}}∥.

Integration of (3.17) with respect to x^{2} yields

(3.18) ∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | L_{2|x^{1}} | ℓ_{∞|k_{2}}^{n_{2}} | ℓ_{∞|k_{1}}^{r} | L_{1,x^{2}}∥

≤ c∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ(n_{1},n_{2}) ; \overline{∞} | L_{u|x^{1}} | L_{1|x^{2}}∥.

Real interpolation with the parameters θ and 1 on both sides of (3.18) gives

(3.19)  ∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | L_{2|x^{1}} | ℓ_{∞|k_{2}}^{n_{2}} | ℓ_{1|k_{1}}^{n_{1}/2} | L_{1,x^{2}}∥

≤ c∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ(n_{1},n_{2}) ; \overline{∞} | L\overline{1}∥

and we see that the inequalities (3.11), (3.15), (3.17) can be combined to obtain

(3.20)  ∥f | L\overline{∞}(ℝ^{n_{1}} × ℝ^{n_{2}})∥ ≤ c∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ(n_{1},n_{2}) ; \overline{∞} | L\overline{1}∥
≤ c∥f | S_{\overline{1},\overline{2}}^{(n_{1},n_{2})}F ∥.

Thus (3.7) is proved.

The inequality (3.8) follows simply by Hölder’s inequality.

The proof (3.9) uses similar arguments as that of (3.7).

Finally, to prove (3.10) we start from

∥f(x^{1}, x^{2}) | L_{(∞,1)}∥

≤ c ∑_{k_{2}=0}^{∞} ∑_{k_{1}=0}^{∞} 2^{−k_{2}n_{2}}∥2^{k_{1}n_{1}/2+k_{2}n_{2}}f_{k_{1},k_{2}}(x^{1}, x^{2}) | L_{2|x^{1}} | L_{1|x^{2}}∥

≤ c∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{1}^{n_{1}/2}(L_{2}) | L_{1}(ℓ_{∞}^{n_{2}})∥,

and then it is enough to use (3.19). Step 3. The following mixed norm estimates hold:

(3.21) ∥f | L_{(q_{1},∞)}∥ ≤ cq_{1}^{1−1/p_{1}}∥f | S_{(p_{1},1),\overline{2}}^{(n_{1}/p_{1},n_{2})}F ∥ if 1 ≤ p_{1} < q_{1} < ∞,
(3.22) ∥f | L_{(∞,q_{2})}∥ ≤ cq_{2}^{1−1/p_{2}}∥f | S_{(1,p_{2}),\overline{2}}^{(n_{1},n_{2}/p_{2})}F ∥ if 2 ≤ p_{2} < q_{2} < ∞,
(3.23) ∥f | L_{(q_{1},1)}∥ ≤ cq_{1}^{1−1/p_{1}}∥f | S_{(p_{1},1),\overline{2}}^{(n_{1}/p_{1},n_{2})}F ∥ if p_{1} ≥ 2,
(3.24) ∥f | L_{(1,q_{2})}∥ ≤ cq_{2}^{1−1/p_{2}}∥f | S_{(1,p_{2}),\overline{2}}^{(n_{1},n_{2}/p_{2})}F ∥ if p_{2} ≥ 2.

We shall briefly go through the proof of (3.21). Similarly as in the proof of Lemma 3.4 (cf. (3.4) and the preceding estimates) we obtain

(3.25) ∥f(x^{1}, x^{2}) | L_{(q_{1},∞)}∥
≤ cq_{1}^{1−1/p_{1}}∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{p_{1}}^{n_{1}/p_{1}}(L_{p_{1}}) | ℓ_{1}^{n_{2}/2}(L_{2})∥

and, analogously as in the proof of (3.15), it is possible to show that (3.25) implies

(3.26) ∥f(x^{1}, x^{2}) | L_{(q_{1},∞)}∥
≤ cq_{1}^{1−1/p_{1}}∥f_{k_{1},k_{2}}(x^{1}, x^{2}) | ℓ_{p_{1}}^{n_{1}/p_{1}}(L_{p_{1}}) | L_{1}(ℓ_{∞}^{n_{2}})∥.

Since p_{1} ≥ 2, (3.26) yields (3.21).

The proof of the remaining estimates (3.22)–(3.24) makes use of a similar technique. Step 4. Now we are in a position to complete the proof of (3.5). By (3.7) and (3.8)–(3.10) we get (3.5) for p_{1} = p_{1} = 1. To get (3.5) for p_{1} = 1 and p_{2} ≥ 2 we use (3.21), (3.23) and the mixed norm interpolation inequality

∥f | L_{(q_{1},q_{2})}∥ ≤ ∥f | L_{(q_{1},∞)}∥^{1−1/q_{2}}∥f | L_{(q_{1},1)}∥^{1/q_{2}}.

If p_{1} ≥ 2 and p_{2} ≥ 2, we proceed analogously, using (3.22), (3.24), and the inequality

∥f | L_{(q_{1},q_{2})}∥ ≤ ∥f | L_{(∞,q_{2})}∥^{1−1/q_{1}}∥f | L_{(1,q_{2})}∥^{1/q_{1}}

to prove (3.5) again. By virtue of our consideration at the beginning of the proof we are done. □

Remark 3.6. The estimate (3.5) cannot be improved with respect to the exponents 1 − 1/p_{j}. It is enough to take f̃(x) = f(x^{1}) ⊗ ⋯ ⊗ f(x^{m}), where f^{j} are the “extremal functions” [ET2, Subsection 2.7.1].

Corollary 3.7. Under the assumptions of Theorem 3.5 the estimate

∥f | L_{q}∥ ≤ cq^{m− ∑_{j=1}^{m} 1/p_{j}}∥f | S\overline{r} ; \overline{p},\overline{2}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥

holds for every q > max_{j=1,...,m} p_{j}.

Of particular interest are the reduced Sobolev spaces described in Sections 1 and studied in Section 2. In this case we have Corollary 3.8. Let M = S(N, n_{1}, …, n_{m}).

(i) If q_{j} > n_{j} (j = 1, …, m), q > max_{j} n_{j}, then

∥f | L\overline{q}∥ ≤ c ∏_{j=1}^{m} q_{j}^{1−1/n_{j}}∥f | W M ; \overline{n}(ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥,

where \overline{n} = (n_{1}, …, n_{m}) and

∥f | L_{q}∥ ≤ cq^{m(1− ∑_{j=1}^{m} 1/n_{j})}∥f | W M ; \overline{n}(ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}})∥.

(ii) If n_{1} = ⋯ = n_{m} = n > m, then

∥f | L_{q}∥ ≤ cq^{m(1−1/n)}∥f | W_{n}^{M}(ℝ^{N})∥.

This is a consequence of Theorem 3.6, Corollary 3.7 and the identity

W_{p}^{M}(ℝ^{N}) = S_{\overline{p},\overline{2}}^{(1,...,1)}F (ℝ^{n_{1}} × ⋯ × ℝ^{n_{m}}),

which follows from the observations following Definition 3.2.

Remark 3.9. Let us illustrate the previous corollary by a typical example. Let

S = {(1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 0, 1)}

and let W_{2}^{S} be the corresponding reduced Sobolev space. Then m = n = 2, thus for q > 2 we have

∥f | L_{q}∥ ≤ cq∥f | W_{2}^{S}∥

in contrast to

∥f | L_{q}∥ ≤ cq^{1/2}∥f | W_{2}^{2}∥.

Both the last imbeddings are sharp.

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