H-Distributions : An Extension of H-Measures to an Lp-Lq Setting

and Applied Analysis 3 can be extended to a continuous mapping Tφ : L d → L d . One calls operator Tφ the L-multiplier operatorwith symbol φ. Theorem 1.4 Hörmander-Mikhlin . Let φ ∈ L∞ d have partial derivatives of order less than or equal to κ, where κ is the least integer strictly greater than d/2 (i.e., κ d/2 1). If, for some constant k > 0, ∀r > 0 ( ∀α ∈ 0 ) |α| ≤ κ ⇒ ∫ r/2≤|ξ|≤r ∣∣ ∣D ξφ ξ ∣∣ ∣ 2 dξ ≤ k2rd−2|α|, 1.6 then, for any p ∈ 〈1,∞〉 and the associated multiplier operator Tφ, there exists a constant Cd (depending only on the dimension d; see [18, page 367]) such that ∥Tφ ∥∥ Lp →Lp ≤ Cdmax { p, 1 p − 1 } ( k ∥φ ∥∥ ∞ ) . 1.7 Remark 1.5. It is important to notice that, according to 19, Section 3.2, Example 2 , if the symbol of a multiplier is aC function defined on the unit sphere Sd−1 ⊆ d , then the constant k from Theorem 1.4 can be taken to be equal to ‖φ‖Cκ Sd−1 . By an application of Theorem 1.4, in Section 2 we are able to introduceH-distributions see Theorem 2.1 —an extension ofH-measures in the L-setting. Its proof is the main result of the paper and forms Section 3. We conclude in Section 4 by an L-variant of the localisation principle and a proof of an L, L ′ -variant of the div-curl lemma. Remark 1.6. Recently, variants of H-measures with a different scaling were introduced the parabolic H-measures 15, 20 and the ultraparabolic H-measures 21 . We can apply the procedure from this paper to extend the notion of such H-measures to the L-setting in the same fashion as it is given here based on Theorem 1.1 for the classicalH-measures. 2. A Generalisation of H-Measures We have already seen Remark 1.2 that an H-measure μ corresponding to a sequence un in L2 d can describe its loss of strong compactness. We would like to introduce a similar notion describing the loss at least in Lloc of strong compactness for sequences weakly converging in L d . Consider a sequence un weakly converging to zero in L d and satisfying the following sequence of differential equations:


Introduction
In the study of partial differential equations, quite often it is of interest to determine whether some L p weakly convergent sequence converges strongly.Various techniques and tools have been developed for that purpose for the state of the art twenty years ago, see 1 ; of more modern ones, we only mention the H-measure of Tartar 2 , independently introduced by Gérard 3 under the name of microlocal defect measures.H-measures proved to be very powerful tool in a number of applications see, e.g., 4-13 and references therein, which is surely an incomplete list .The main theorem on the existence of H-measures, in an equivalent form suitable for our purposes, reads as following: Theorem 1.1.If scalar sequences u n , v n 0 weakly in L 2 Ê d , then there exist subsequences u n , v n and a complex Radon measure μ on Ê d × S d−1 such that, for every ϕ 1 , ϕ 2 ∈ C 0 Ê d and every ψ ∈ C S d−1 , where A ψ is the Fourier multiplier operator with the symbol ψ: We call the measure μ the H-measure corresponding to the sequence u n , v n .In fact, it corresponds to the nondiagonal element of the corresponding 2 × 2 matrix Radon measure of the vector function u n , v n cf.14 .
Remark 1.2.After applying the Plancherel theorem, the term under the limit sign in Theorem 1.1 takes the form where by u ξ Fu ξ Ê d e −2πix•ξ u x dx we denote the Fourier transform on Ê d with the inverse Fv x : Ê d e 2πix•ξ v ξ dξ .Of course, ψ is extended by homogeneity i.e., ψ ξ : In the particular case of u n v n , μ roughly describes the loss of strong L 2 precompactness of sequence u n .Indeed, it is not difficult to see that if u n is strongly convergent in L 2 , then the corresponding H-measure is trivial; on the other hand, if the H-measure is trivial, then u n → 0 in L 2 loc Ê d for the details in a similar situation see, 15 .In order to explain how to apply this idea to L p -weakly converging sequences when p / 2, consider the integral in 1.1 .The Cauchy-Schwartz inequality and the Plancherel theorem imply see, e.g., 2, page 198 where C depends on a uniform bound for u n , v n L 2 Ê d ;Ê 2 .In essence, this fact and the linearity of integral in 1.1 with respect to ϕ 1 ϕ 2 and ψ enable us to state that the limit in 1.1 is a Radon measure a bounded linear functional on In 3 , the question whether it is possible to extend the notion of H-measures or microlocal defect measures in the terminology used there to the L p framework is posed see also 16, page 331 .We will consider only the case p ∈ 1, ∞ i.e., 1 < p < ∞; its dual exponent we will consistently denote by p .
To answer that question, one necessarily needs precise bounds for the Fourier multiplier operator A ψ as a mapping from L p Ê d to L p Ê d .The bounds are given by the famous H örmander-Mikhlin theorem 17, 18 .
then, for any p ∈ 1, ∞ and the associated multiplier operator T φ , there exists a constant C d (depending only on the dimension d; see [18, page 367]) such that Remark 1.5.It is important to notice that, according to 19, Section 3.2, Example 2 , if the symbol of a multiplier is a C κ function defined on the unit sphere S d−1 ⊆ Ê d , then the constant k from Theorem 1.4 can be taken to be equal to φ C κ S d−1 .By an application of Theorem 1.4, in Section 2 we are able to introduce H-distributions see Theorem 2.1 -an extension of H-measures in the L p -setting.Its proof is the main result of the paper and forms Section 3. We conclude in Section 4 by an L p -variant of the localisation principle and a proof of an L p , L p -variant of the div-curl lemma.
Remark 1.6.Recently, variants of H-measures with a different scaling were introduced the parabolic H-measures 15, 20 and the ultraparabolic H-measures 21 .We can apply the procedure from this paper to extend the notion of such H-measures to the L p -setting in the same fashion as it is given here based on Theorem 1.1 for the classical H-measures.

A Generalisation of H-Measures
We have already seen Remark 1.2 that an H-measure μ corresponding to a sequence u n in L 2 Ê d can describe its loss of strong compactness.We would like to introduce a similar notion describing the loss at least in L 1 loc of strong compactness for sequences weakly converging in L p Ê d .Consider a sequence u n weakly converging to zero in L p Ê d and satisfying the following sequence of differential equations: where If we instead take u n ∈ L p Ê d , for p < 2, then we cannot apply the same tool.Here we propose the following replacement.
for some q ≥ max{p , 2}, then there exist subsequences u n , v n and a complex valued distribution μ ∈ D Ê d × S d−1 of order not more than

2.2
where We call the functional μ the H-distribution corresponding to a subsequence of u n and v n .Of course, for q ∈ 1, ∞ , the weak * convergence coincides with the weak convergence.
If we are given sequences u n and v n defined on an open set Ω ⊆ Ê d , then we can extend the functions by zero to Ê d , preserving the convergence, and then apply Theorem 2.1 in the above form.The resulting H-distribution will be supported on Ð Ω × S d−1 , as it can easily be seen by taking test functions ϕ 1 and ϕ 2 supported within the complement of the closure Ð Ω.
Remark 2.2.Notice that, unlike what was the case with H-measures, it is not possible to write 2.2 in a form similar to 1.3 since, according to the Hausdorff-Young inequality, This means that we are not able to estimate F ϕ 2 v n L q Ê d , for q > 2, which would appear from 2.2 when rewriting it in a form similar to 1.3 .
Remark 2.3.In Theorem 2.1, we clearly distinguish between u n ∈ L p Ê d and v n ∈ L q Ê d .For p ≥ 2, p ≤ 2 and we can take q ≥ 2; in particular, this covers the classical L 2 case including u n v n .Even more, in this case p ≥ 2 , the assumptions of Theorem 2.1 imply that u n , v n 0 in L 2 loc Ê d and we can again use a classical framework, resulting in a distribution μ of order zero a Radon measure, not necessary bounded , instead of a more general distribution of order κ.The real improvement in Theorem 2.1 is for the case p < 2.
Remark 2.4.For applications, it might be of interest to extend the result to vector-valued functions.In the case when Ù n ∈ L p Ê d k and Ú n ∈ L q Ê d l , the result is a matrix valued distribution μ μ ij , where i ∈ 1, . . ., t and j ∈ 1, . . ., l.It should be noted that, in contrast to what is done with H-measures, in general we cannot consider H-distributions corresponding to the same sequence, but only to a pair of sequences, and H-distribution would correspond to nondiagonal blocks for H-measures 14 see also the example at the beginning of Section 4 .

Proof of Theorem 2.1
In order to prove the theorem, we need a consequence of Tartar's first commutation lemma 2, Lemma 1.7 .First, for ψ ∈ C κ S d−1 and b ∈ C 0 Ê d , define the Fourier multiplier operator A ψ and the operator of multiplication B on L p Ê d , by the formulae

3.1
Notice that ψ satisfies the conditions of the H örmander-Mikhlin theorem see Remark 1.5 .
Therefore, A ψ and B are bounded operators on L p Ê d , for any p ∈ 1, ∞ .We are interested in the properties of their commutator, C A ψ B − BA ψ .
Lemma 3.1.Let v n be bounded in both L 2 Ê d and L r Ê d , for some r ∈ 2, ∞ , and such that v n 0 in the sense of distributions.Then the sequence Cv n strongly converges to zero in L q Ê d , for any q ∈ 2, r \ {∞}.
Proof.If r < ∞, then we can apply the classical interpolation inequality: for α ∈ 0, 1 such that 1/q α/2 1 − α /r.As C is a compact operator on L 2 Ê d by the first commutation lemma, while C is bounded on L r Ê d , from 3.2 we get the claim.
In the case r ∞, notice that we do not have the boundedness of A ψ on L ∞ , but only on L p , for p < ∞.Therefore, we take p ∈ q, ∞ and by the interpolation inequality conclude that v n is bounded in L p .Now, we can proceed as above, with r replaced by p.
Proof of Theorem 2.1.The first equality from 2.2 follows from the fact that the adjoint operator A * ψ corresponding to A ψ is actually the multiplier operator A ψ see 17, Theorem 7.4.3 .This means that we take the duality product to be sesquilinear, i.e., antilinear in the second variable, in order to get the scalar product when p p 2 which is exactly what we need.We can now concentrate our attention on the second equality in 2.2 .Since u n to the H örmander-Mikhlin theorem for any We can write Ê d l∈AE K l , where K l form an increasing family of compact sets e.g., closed balls around the origin of radius l ; therefore supp ϕ 2 ⊆ K l for some l ∈ AE.One has where χ l is the characteristic function of K l .In the second equality, one has used Lemma 3.1.This allows us to express the above integrals as bilinear functionals, after denoting ϕ ϕ 1 ϕ 2 : , as according to the H ölder inequality, Theorem 1.4 and Remark 1.5: where the constant C depends on L p K l -norm and L p K l -norm of the sequences u n and v n , respectively.
For each l ∈ AE, we can apply Lemma 3.2 actually, the operators are defined in its proof to obtain operators B l ∈ L C K l Ê d ; C κ S d−1 .Furthermore, for the construction of B l , we can start with a defining subsequence for B l−1 , so that the convergence will remain valid on C K l−1 Ê d , in such a way obtaining that B l is an extension of B l−1 .This allows us to define the operator B on C c Ê d : for, ϕ ∈ C c Ê d , we take l ∈ AE such that supp ϕ ⊆ K l , and set Bϕ : B l ϕ.Because of the above-mentioned extension property, this definition is good, and one has a bounded operator:

3.8
In such a way one has got a bounded linear operator B on the space C c Ê d equipped with the uniform norm; the operator can be extended to its completion, the Banach space Now, we can define μ ϕ, ψ : Bϕ, ψ , which satisfies 2. We conclude this section by a simple lemma and its proof, which was used in the proof of Theorem 2.1.Lemma 3.2.Let E and F be separable Banach spaces and b n an equibounded sequence of bilinear forms on E × F (more precisely, there is a constant C such that, for each n ∈ AE one has Then, there exists a subsequence b n k and a bilinear form b (with the same bound C) such that Proof.To each b n , we associate a bounded linear operator B n : E → F by The above expression clearly defines a function i.e., B n ϕ ∈ F is uniquely determined ; it is linear in ϕ and bounded:

3.11
Let G ⊆ E be a countable dense subset; for each ϕ ∈ G, the sequence B n ϕ is bounded in F , so by the Banach theorem there is a subsequence such that

3.12
By repeating this construction countably many times and then applying the Cantor diagonal procedure, we get a subsequence Then, it is standard to extend B to a bounded linear operator on the whole space E. 3.14

Some Applications
It is well-known that weak convergences are ill behaved under nonlinear transformations in contrast to their good behaviour under linear transformations .Only in some particular cases of compensation, it is even possible to pass to the limit in a product of two weakly converging sequences.
The prototype of this compensation effect is Tartar-Murat's div-curl lemma cf.24, Theorem 7.1 .
For simplicity, consider two vector-valued sequences, u 1 n , u 2 n and v 1 n , v 2 n , converging to zero weakly in n are both contained in a compact set of H −1 loc Ê 2 which then implies that they converge to zero strongly in H −1 loc Ê 2 .We can define Í n : Ù n Ú n , which on a subsequence defines a 4 × 4 H-measure μ.By the localisation principle 2, Theorem 1.6 and 14, Theorem 2 , as the above relations can be written in the form A 1 , A 2 are 4 × 4 constant matrices with all entries zero except 0 weak * in the sense of Radon measures and therefore in the sense of distributions as well .
For the above, one has used only the nondiagonal blocks to products of u i n and v j n ; in fact, the calculation shows that μ 11 12 μ 22   12   0, which gives the above result.
In order to get a similar result using H-distributions, we first show that the following localisation principle holds.
Take an arbitrary sequence v n bounded in L ∞ Ê d , and by μ denote the H-distribution corresponding to some subsequences of sequences u n and v n .Then, in the sense of distributions on Ê d × S d−1 , the function x, ξ → d i 1 A i x ξ i being the symbol of the linear partial differential operator with C κ 0 coefficients.
Proof.In order to prove the theorem, we need a particular multiplier, the so called Marcel Riesz potential I 1 : A |2πξ| −1 , and the Riesz transforms R j : A ξ j /i|ξ| 19, V.1,2 .We note that id.,V.2.3

4.5
From here, using the density argument and the fact that R j is bounded from L p Ê d to itself, we conclude that We should prove that the H-distribution corresponding to the chosen subsequences of u n and v n satisfies 4.4 .To this end, take the following sequence of test functions: where Then, apply the right-hand side of 4.3 , which converges strongly to 0 in W −1,q Ê d by the assumption, to a weakly converging sequence φ n in the dual space W 1,q Ê d .
We can do that since φ n is a bounded sequence in W 1,r Ê d for any r ∈ 1, ∞ .Indeed, A ψ ϕ 2 v n is bounded in any L r Ê d r > 1 .By the well-known fact 19, Theorem V.1 that I 1 is bounded from L q Ê d to L q * Ê d , for q ∈ 1, d and 1/q * 1/q−1/d, φ n is bounded in L q * Ê d for all sufficiently large q * .Then, take q * ≥ r and due to the compact support of ϕ 1 , one has that L q * boundedness implies the same in L r .On the other hand, R j is bounded from L r Ê d to itself, for any r ∈ 1, ∞ , thus; Therefore, one has the sequence is bounded and 0 is the only accumulation point, so the whole sequence converges to 0 4.7 Concerning the left-hand side of 4.3 , according to 4.5 , one has

4.8
The first term on the right is of the form of the right-hand side of 2.2 .The integrand in the second term is supported in a fixed compact and weakly converging to 0 in L p , so strongly in W −1,r , where r is such that p r * i.e., r dp/ d − p .Of course, the argument giving the boundedness of φ n in W 1,q Ê d above applies also to r instead of q .
Remark 4.2.Notice that the assumption of the strong convergence of f n in W −1,q Ê d can be relaxed to local convergence, because in the proof we used a cutoff function ϕ 1 .
Let us return to the simple example from the beginning of this section; consider two vector-valued sequences u 1 n , u 2 n and v 1 n , v 2 n , this time converging to zero weakly in L p Ê and L p Ê , respectively.Assume that the sequence ∂ and by weak * compactness it has a weakly converging subsequence.However, we can say more-the whole sequence converges to zero.Denote by μ ij the H-distribution corresponding to some sub sequences of u are weakly precompact, while the only possible limit is zero, so Now, from the compactness properties of the Riesz potential I 1 see the proof of previous theorem , we conclude that, for every ϕ ∈ C c Ê 2 and ψ ∈ C κ S d−1 , the following limit holds in L p Ê 2 : After inserting ψ ≡ 1 in the definition of H-distribution 2.2 , we immediately reach the conclusion.This proof is similar to the L 2 case, but it should be noted that there we had used only a nondiagonal block of 4 × 4 H-measure, which corresponds to the only available 2 × 2 H-distribution.
There is no reason to limit oneself to two dimensions; take Ù n and Ú n converging weakly to zero in L p Ê d d and L p Ê d d , and by μ denote d × d matrix H-distribution corresponding to some chosen subsequences of Ù n and Ú n .
From Ú Ù n → 0 strongly in W −1,p loc Ê d , for ϕ 1 ∈ C c Ê d and ψ ∈ C κ S d−1 , one has as in 4.10 that

4.16
After forming a product with ϕ 2 v n , integrating and passing to the limit, we conclude that ξ Ì μ ¼, 4.17 namely, that the columns of μ are perpendicular to ξ.
On the other hand, from ÙÖÐ Ú n → 0 strongly in W −1,p loc Ê d d×d , in an analogous way, we conclude that, for each row denoted by μ i of μ, for all j, k, one has ξ j μ ik − ξ k μ ij 0, 4.18 so the rows of μ are proportional to ξ and μ λ ⊗ ξ a rank-one matrix , λ i being the constants of proportionality.So, the columns of μ are proportional to λ, while earlier we showed that they are perpendicular to ξ.Thus, ØÖ μ λ • ξ 0, which implies the convergence Ù n • Ú n 0, as in the two-dimensional situation.
The above result is the well-known Murat's div-curl lemma in the L p , L p -setting 24, 25 , which we state as a theorem.

2 .
We can restrict B to an operator B defined only on C ∞ c Ê d ; as the topology on C ∞ c Ê d is stronger than the one inherited from C 0 Ê d , the restriction remains continuous.Furthermore, C κ S d−1 is the space of distributions of order κ, which is a subspace of D S d−1 .In such a way, one has a continuous operator from C ∞ c Ê d to D S d−1 , which by the Schwartz kernel theorem can be identified to a distribution from D Ê d × S d−1 for details cf.23, Chapter VI .

Theorem 4 . 3 .
Let Ù n and Ú n be vector-valued sequences converging to zero weakly in L p Ê d d and L p Ê d d , respectively.Assume that the sequence Ú Ù n is bounded in L p Ê d and the sequenceÙÖÐ Ú n is bounded in L p Ê d d×d .Then, the sequence Ù n • Ú n converges to zero in the sense of distributions (or vaguely in the sense of Radon measures).
f n → 0 strongly in the Sobolev space H −1 Ê d .When dealing with the latter equation, it is standard to multiply 2.1 by A ψ/|ξ| φu n , for φ ∈ C 0 Ê d, where A ψ/|ξ| is the multiplier operator with symbol ψ ξ/|ξ| /|ξ|, ψ ∈ C S d−1 , and then pass to the limit see, e.g., 14, 22 .If u n ∈ L 2 Ê d , then we can apply the classical H-measures to describe the defect of compactness for u n .