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

Authors: Nenad Antonić; Darko Mitrović
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We use the continuity of Fourier multiplier operators onLp to introduce the H-distributions—an extension of H-measures in the Lp framework. We apply the H-distributions to obtain an Lp version of the localisation principle and reprove the Murat Lp-Lp′ variant of div-curl lemma.


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## Body

## 1. Introduction

In the study of partial differential equations, quite often it is of interest to determine whether someLp 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 ofmicrolocal 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 sequencesun,vn⇀0 weakly in L2(ℝd),  then there exist subsequences (un′),(vn′) and a complex Radon measure μ on ℝd×Sd-1 such that, for every φ1,φ2∈C0(ℝd) and every ψ∈C(Sd-1),
(1.1)lim⁡n′〈Aψ(φ1un′)∣φ2vn′〉=lim⁡n′∫RdAψ(φ1un′)φ2vn′̅dx=〈μ,φ1φ̅2ψ〉,
where 𝒜ψ is the Fourier multiplier operator with the symbol ψ:
(1.2)Aψu:=F̅(ψû).We call the measureμ theH-measure corresponding to the sequence (un,vn). In fact, it corresponds to the nondiagonal element of the corresponding 2×2 matrix Radon measure of the vector function (un,vn) (cf. [14]).Remark 1.2.
After applying the Plancherel theorem, the term under the limit sign in Theorem1.1 takes the form
(1.3)∫Rdφ1un′̂φ2vn′̂̅ψdξ,
where by û(ξ)=(ℱu)(ξ)=∫ℝde-2πix·ξu(x)dx we denote the Fourier transform on ℝd (with the inverse (ℱ̅v)(x):=∫ℝde2πix·ξv(ξ)dξ). Of course, ψ is extended by homogeneity (i.e., ψ(ξ):=ψ(ξ/|ξ|)) to ℝd∖{0}.
In the particular case ofun=vn, μ roughly describes the loss of strong L2 precompactness of sequence (un). Indeed, it is not difficult to see that if (un) is strongly convergent in L2, then the corresponding H-measure is trivial; on the other hand, if the H-measure is trivial, then un→0 in Lloc2(ℝd) (for the details in a similar situation see, [15]).In order to explain how to apply this idea toLp-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]) (1.4)|∫RdAψ(φ1un′)φ2vn′¯dx|≤C‖ψ‖C(Sd-1)‖φ1φ2¯‖C0(Rd),
where C depends on a uniform bound for ∥(un,vn)∥L2(ℝd;ℝ2). In essence, this fact and the linearity of integral in (1.1) with respect to φ1φ2¯ and ψ enable us to state that the limit in (1.1) is a Radon measure (a bounded linear functional on C0(ℝd×Sd-1)). Furthermore, the bound is obtained by a simple estimate ∥𝒜ψ∥L2→L2≤∥ψ∥L∞(ℝd) and the fact that (un,vn) is a bounded sequence in L2(ℝd;ℝ2).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 Lp 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 operator𝒜ψ as a mapping from Lp(ℝd) to Lp(ℝd). The bounds are given by the famous Hörmander-Mikhlin theorem [17, 18].Definition 1.3.
Letϕ:ℝd→ℂ satisfy (1+|x|2)-k/2ϕ∈L1(ℝd) for some k∈ℕ0.Then, ϕ is called the Fourier multiplier on Lp(ℝd),if ℱ̅(ϕℱ(θ))∈Lp(ℝd) for any θ∈𝒮(ℝd), and
(1.5)S(Rd)∋θ⟼F̅(ϕF(θ))∈Lp(Rd)
can be extended to a continuous mapping Tϕ:Lp(ℝd)→Lp(ℝd). One calls operator Tϕ theLp-multiplier operator with symbolϕ.Theorem 1.4 (Hörmander-Mikhlin).
Letϕ∈L∞(ℝd) 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,
(1.6)(∀r>0)(∀α∈N0d)|α|≤κ⟹∫r/2≤|ξ|≤r|Dξαϕ(ξ)|2dξ≤k2rd-2|α|,
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
(1.7)‖Tϕ‖Lp→Lp≤Cdmax⁡{p,1p-1}(k+‖ϕ‖∞).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 Cκ function defined on the unit sphere Sd-1⊆ℝd, then the constant k from Theorem 1.4 can be taken to be equal to ∥ϕ∥Cκ(Sd-1).
By an application of Theorem1.4, in Section 2 we are able to introduce H-distributions (see Theorem 2.1)—an extension of H-measures in the Lp-setting. Its proof is the main result of the paper and forms Section 3. We conclude in Section 4 by an Lp-variant of the localisation principle and a proof of an (Lp,Lp′)-variant of the div-curl lemma.Remark 1.6.
Recently, variants ofH-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 Lp-setting in the same fashion as it is given here based on Theorem 1.1 for the classical H-measures.


## 2. A Generalisation ofH-Measures

We have already seen (Remark1.2) that an H-measure μ corresponding to a sequence (un) in L2(ℝd) can describe its loss of strong compactness. We would like to introduce a similar notion describing the loss (at least in Lloc1) of strong compactness for sequences weakly converging in Lp(ℝd).Consider a sequence(un) weakly converging to zero in Lp(ℝd) and satisfying the following sequence of differential equations: (2.1)∑i=1d∂i(Ai(x)un(x))=fn(x),
where Ai∈C0(ℝd) and fn→0 strongly in the Sobolev space H-1(ℝd). When dealing with the latter equation, it is standard to multiply (2.1) by 𝒜ψ/|ξ|(ϕun), for ϕ∈C0(ℝd), where 𝒜ψ/|ξ| is the multiplier operator with symbol ψ(ξ/|ξ|)/|ξ|, ψ∈C(Sd-1), and then pass to the limit (see, e.g., [14, 22]). If un∈L2(ℝd), then we can apply the classical H-measures to describe the defect of compactness for (un).If we instead takeun∈Lp(ℝd), for p<2, then we cannot apply the same tool. Here we propose the following replacement.Theorem 2.1.
Ifun⇀0 in  Lp(ℝd) and vn⇀*v in  Lq(ℝd) for some q≥max{p′,2},  then there exist subsequences (un′),  (vn′) and a complex valued distribution μ∈𝒟′(ℝd×Sd-1) of order not more than κ=[d/2]+1 in ξ, such that, for every φ1,φ2∈Cc∞(ℝd) and ψ∈Cκ(Sd-1) one has:
(2.2)lim⁡n′→∞∫RdAψ(φ1un′)(x)(φ2vn′)(x)¯dx=lim⁡n′→∞∫Rd(φ1un′)(x)Aψ¯(φ2vn′)(x)¯dx=〈μ,φ1φ¯2ψ〉,
where 𝒜ψ:Lp(ℝd)→Lp(ℝd) is a multiplier operator with symbol ψ∈Cκ(Sd-1).We call the functionalμ the H-distribution corresponding to (a subsequence of) (un) and (vn). Of course, for q∈〈1,∞〉, the weak * convergence coincides with the weak convergence.If we are given sequences(un) and (vn) defined on an open set Ω⊆ℝd, then we can extend the functions by zero to ℝd, preserving the convergence, and then apply Theorem 2.1 in the above form. The resulting H-distribution will be supported on ClΩ×Sd-1, as it can easily be seen by taking test functions φ1 and φ2 supported within the complement of the closure ClΩ.Remark 2.2.
Notice that, unlike what was the case withH-measures, it is not possible to write (2.2) in a form similar to (1.3) since, according to the Hausdorff-Young inequality, ∥ℱ(u)∥Lp′(ℝd)≤C∥u∥Lp(ℝd) only if 1<p≤2. This means that we are not able to estimate ∥ℱ(φ2vn)∥Lq′(ℝd), for q>2, which would appear from (2.2) when rewriting it in a form similar to (1.3).Remark 2.3.
In Theorem2.1, we clearly distinguish between un∈Lp(ℝd) and vn∈Lq(ℝd). For p≥2,p′≤2 and we can take q≥2;in particular, this covers the classical L2 case (including un=vn). Even more, in this case (p≥2), the assumptions of Theorem 2.1 imply that un,vn⇀0 in Lloc2(ℝd) 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 whenun∈Lp(ℝd)k and vn∈Lq(ℝd)l, the result is a matrix valued distributionμ=[μij],where i∈1,…,t and j∈1,…,l.
It should be noted that, in contrast to what is done withH-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).


## 3. Proof of Theorem2.1

In order to prove the theorem, we need a consequence of Tartar's first commutation lemma [2, Lemma  1.7]. First, for ψ∈Cκ(Sd-1) and b∈C0(ℝd), define the Fourier multiplier operator 𝒜ψ and the operator of multiplication B on Lp(ℝd), by the formulae (3.1)F(Aψu)(ξ)=ψ(ξ|ξ|)F(u)(ξ),Bu(x)=b(x)u(x).
Notice that ψ satisfies the conditions of the Hörmander-Mikhlin theorem (see Remark 1.5). Therefore, 𝒜ψ and B are bounded operators on Lp(ℝd), for any p∈〈1,∞〉. We are interested in the properties of their commutator, C=𝒜ψB-B𝒜ψ.Lemma 3.1.
Let(vn) be bounded in both L2(ℝd) and Lr(ℝd), for some r∈〈2,∞], and such that vn⇀0 in the sense of distributions. Then the sequence (Cvn) strongly converges to zero in Lq(ℝd), for any q∈[2,r]∖{∞}.Proof.
Ifr<∞, then we can apply the classical interpolation inequality:
(3.2)‖Cvn‖q≤‖Cvn‖2α‖Cvn‖r1-α,
for α∈〈0,1〉 such that 1/q=α/2+(1-α)/r. As C is a compact operator on L2(ℝd) by the first commutation lemma, while C is bounded on Lr(ℝd), from (3.2) we get the claim.
In the caser=∞, notice that we do not have the boundedness of 𝒜ψ on L∞, but only on Lp, for p<∞. Therefore, we take p∈〈q,∞〉 and by the interpolation inequality conclude that (vn) is bounded in Lp. Now, we can proceed as above, with r replaced by p.Proof of Theorem2.1.
The first equality from (2.2) follows from the fact that the adjoint operator 𝒜ψ* corresponding to 𝒜ψ is actually the multiplier operator 𝒜ψ̅ (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)
(3.3)〈Aψ(φ1un′),φ2vn′〉LpLp′=〈φ1un′,Aψ¯(φ2vn′)〉Lp′Lp,
which is exactly what we need. We can now concentrate our attention on the second equality in (2.2).
Sinceun⇀0 in Lp(ℝd), while for v∈Lq(ℝd) one has φ1𝒜ψ̅(φ2v)¯∈Lp′(ℝd), according to the Hörmander-Mikhlin theorem for any φ1,φ2∈Cc∞(ℝd) and ψ∈Cκ(Sd-1), it follows that
(3.4)lim⁡n→∞∫Rdφ1unAψ¯(φ2v)¯dx=0.
We can writeℝd=⋃l∈ℕKl, where Kl form an increasing family of compact sets (e.g., closed balls around the origin of radius l); therefore suppφ2⊆Kl for some l∈ℕ. One has
(3.5)lim⁡n→∞∫Rdφ1unAψ¯(φ2vn)̅dx=lim⁡n→∞∫Rdφ1unAψ̅[φ2χl(vn-v)]¯dx=lim⁡n→∞∫Rdφ1φ¯2unAψ̅(χl(vn-v))¯dx=lim⁡n→∞∫Rdφ1φ¯2unAψ̅(χlvn)¯dx,
where χl is the characteristic function of Kl. In the second equality, one has used Lemma 3.1.
This allows us to express the above integrals as bilinear functionals, after denotingφ=φ1φ¯2:
(3.6)μn,l(φ,ψ)=∫RdφunAψ̅(χlvn)¯dx.
Furthermore,μn,l is bounded by C̃∥φ∥C0(ℝd)∥ψ∥Cκ(Sd-1), as according to the Hölder inequality, Theorem 1.4 and Remark 1.5:
(3.7)|μn,l(φ,ψ)|≤‖φun‖p‖Aψ̅(χlvn)‖p′≤C̃‖ψ‖Cκ(Sd-1)‖φ‖C0(Rd),
where the constant C̃ depends on Lp(Kl)-norm and Lp′(Kl)-norm of the sequences (un) and (vn), respectively.
For eachl∈ℕ, we can apply Lemma 3.2 (actually, the operators are defined in its proof) to obtain operators Bl∈ℒ(CKl(ℝd);(Cκ(Sd-1))′). Furthermore, for the construction of Bl, we can start with a defining subsequence for Bl-1, so that the convergence will remain valid on CKl-1(ℝd), in such a way obtaining that Bl is an extension of Bl-1.
This allows us to define the operatorB on Cc(ℝd): for, φ∈Cc(ℝd), we take l∈ℕ such that suppφ⊆Kl, and set Bφ:=Blφ. Because of the above-mentioned extension property, this definition is good, and one has a bounded operator:
(3.8)‖Bφ‖(Cκ(Sd-1))′≤C̃‖φ‖C0(Rd).
In such a way one has got a bounded linear operatorB on the space Cc(ℝd) equipped with the uniform norm; the operator can be extended to its completion, the Banach space C0(ℝd).
Now, we can defineμ(φ,ψ):=〈Bφ,ψ〉, which satisfies (2.2).
We can restrictB to an operator B̃ defined only on Cc∞(ℝd); as the topology on Cc∞(ℝd) is stronger than the one inherited from C0(ℝd), the restriction remains continuous. Furthermore, (Cκ(Sd-1))′ is the space of distributions of order κ, which is a subspace of 𝒟′(Sd-1). In such a way, one has a continuous operator from Cc∞(ℝd) to 𝒟′(Sd-1), which by the Schwartz kernel theorem can be identified to a distribution from 𝒟′(ℝd×Sd-1) (for details cf. [23, Chapter VI]).We conclude this section by a simple lemma and its proof, which was used in the proof of Theorem2.1.Lemma 3.2.
LetE and F be separable Banach spaces and (bn) an equibounded sequence of bilinear forms on E×F (more precisely, there is a constant C such that, for each n∈ℕ one has |bn(φ,ψ)|≤C∥φ∥E∥ψ∥F).
Then, there exists a subsequence(bnk) and a bilinear form b (with the same bound C) such that
(3.9)(∀φ∈E)(∀ψ∈F)lim⁡kbnk(φ,ψ)=b(φ,ψ).Proof.
To eachbn, we associate a bounded linear operator Bn:E→F′ by
(3.10)F′〈Bnφ,ψ〉F:=bn(φ,ψ).
The above expression clearly defines a function (i.e., Bnφ∈F′ is uniquely determined); it is linear in φ and bounded:
(3.11)‖Bnφ‖F′=sup⁡ψ≠0⁡|bn(φ,ψ)|‖ψ‖F≤C‖φ‖E.
Let 𝒢⊆E be a countable dense subset; for each φ∈𝒢, the sequence (Bnφ) is bounded in F′, so by the Banach theorem there is a subsequence such that
(3.12)Bn1φ⇀*β1=:B(φ).
By repeating this construction countably many times and then applying the Cantor diagonal procedure, we get a subsequence
(3.13)(∀φ∈G)Bnkφ⇀*B(φ),
such that ∥B(φ)∥F′≤C∥φ∥E.
Then, it is standard to extendB to a bounded linear operator on the whole space E. Clearly,
(3.14)b(φ,ψ):=〈Bφ,ψ〉F′F=lim⁡k〈Bnkφ,ψ〉FF′=lim⁡kbnk(φ,ψ).


## 4. 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,(un1,un2) and (vn1,vn2), converging to zero weakly in L2(ℝ2), such that (∂xun1+∂yun2) and (∂yvn1-∂xvn2) are both contained in a compact set of Hloc-1(ℝ2) (which then implies that they converge to zero strongly in Hloc-1(ℝ2)).We can defineUn:=[unvn], 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 (A1,A2 are 4×4 constant matrices with all entries zero except A111=A122=A332=1 and A341=-1) (4.1)A1∂1Un+A2∂2Un⟶0stronglyinHloc⁡-1(R2)4,
the corresponding H-measure satisfies (ξ1A1+ξ2A2)μ=0. After straightforward calculations this shows that un1vn1+un2vn2⇀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 blocksμ12=μ21* of (4.2)μ=[μ11μ12μ21μ22],
corresponding to products of uni and vnj; in fact, the calculation shows that μ1211+μ1222=0, which gives the above result.In order to get a similar result usingH-distributions, we first show that the following localisation principle holds.Theorem 4.1.
Assume thatun⇀0 in Lp(ℝd) and fn→0 in W-1,q(ℝd), for some q∈〈1,d〉, such that they satisfy
(4.3)∑i=1d∂i(Ai(x)un(x))=fn(x).
Take an arbitrary sequence(vn) bounded in L∞(ℝd), and by μ denote the H-distribution corresponding to some subsequences of sequences (un) and (vn). Then,
(4.4)∑i=1dAi(x)ξiμ(x,ξ)=0
in the sense of distributions on ℝd×Sd-1, the function (x,ξ)↦∑i=1dAi(x)ξi being the symbol of the linear partial differential operator with C0κ coefficients.Proof.
In order to prove the theorem, we need a particular multiplier, the so called (Marcel) Riesz potentialI1:=𝒜|2πξ|-1, and the Riesz transforms Rj:=𝒜ξj/i|ξ| [19, V.1,2]. We note that [id.,V.2.3]
(4.5)∫I1(ϕ)∂jg=∫(Rjϕ)g,g∈S(Rd).
From here, using the density argument and the fact that Rj is bounded from Lp(ℝd) to itself, we conclude that ∂jI1(ϕ)=-Rj(ϕ),forϕ∈Lp(ℝd).
We should prove that theH-distribution corresponding to (the chosen subsequences of) (un) and (vn) satisfies (4.4). To this end, take the following sequence of test functions:
(4.6)ϕn:=φ1(I1∘Aψ(ξ/|ξ|))(φ2vn),
where φ1,φ2∈Cc∞(ℝd) and ψ∈Cκ(Sd-1), κ=[d/2]+1. Then, apply the right-hand side of (4.3), which converges strongly to 0 in W-1,q(ℝd) by the assumption, to a weakly converging sequence (ϕn) in the dual space W1,q′(ℝd).
We can do that since(ϕn) is a bounded sequence in W1,r(ℝd)foranyr∈〈1,∞〉.
Indeed,𝒜ψ(φ2vn) is bounded in any Lr(ℝd) (r>1). By the well-known fact [19, Theorem  V.1] that I1 is bounded from Lq(ℝd)toLq*(ℝd),forq∈〈1,d〉and1/q*=1/q-1/d, ϕn is bounded in Lq*(ℝd) for all sufficiently large q*. Then, take q*≥r and due to the compact support of φ1, one has that Lq* boundedness implies the same in Lr. On the other hand, Rj is bounded from Lr(ℝd) to itself, for any r∈〈1,∞〉, thus; ∂j(φ1(I1∘𝒜ψ(ξ/|ξ|))(φ2vn)) is bounded in Lr(ℝd).
Therefore, one has (the sequence is bounded and 0 is the only accumulation point, so the whole sequence converges to 0)(4.7)〈fn,ϕn〉lim⁡n→∞W-1,q(IRd)W1,q′(Rd)=0.
Concerning the left-hand side of (4.3), according to (4.5), one has
(4.8)〈∑j=1d∂j(Ajun),ϕn〉W1,q′(Rd)W-1,q(Rd)=∫Rd∑j=1dφ¯1AjunA(ξj/|ξ|)ψ(ξ/|ξ|)(φ2un)¯dx-∫Rd∂jφ¯1∑j=1dAjun(I1∘Aψ(ξ/|ξ|))(φ2vn)¯dx.
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 Lp, 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 ϕn in W1,q′(ℝd) above applies also to r instead of q′.
Therefore, from (4.7) and (4.8), one concludes (4.4).Remark 4.2.
Notice that the assumption of the strong convergence offn in W-1,q(ℝd) can be relaxed to local convergence, because in the proof we used a cutoff function φ1.
Let us return to the simple example from the beginning of this section; consider two vector-valued sequences(un1,un2) and (vn1,vn2), this time converging to zero weakly in Lp(ℝ) and Lp′(ℝ), respectively. Assume that the sequence (∂1un1+∂2un2) is bounded in Lp(ℝ2), and (∂2vn1-∂1vn2) in Lp′(ℝ2) (thus precompact in Wloc-1,p(ℝ2), and Wloc-1,p′(ℝ2), resp.).
Then, the sequence(un1vn1+un2vn2) is bounded in L1(ℝ2) so also in ℳb (Radon measures) and by weak * compactness it has a weakly converging subsequence. However, we can say more—the whole sequence converges to zero.
Denote byμij the H-distribution corresponding to (some sub)sequences (of) (un1,un2) and (vn1,vn2).
Since(∂1un1+∂2un2) is bounded in Lp(ℝ2), and (∂2vn1-∂1vn2) is bounded in Lp′(ℝ2), they are weakly precompact, while the only possible limit is zero, so
(4.9)∂1un1+∂2un2⇀0inLp,∂2vn1-∂1vn2⇀0inLp′.
Now, from the compactness properties of the Riesz potential I1 (see the proof of previous theorem), we conclude that, for every φ∈Cc(ℝ2) and ψ∈Cκ(Sd-1), the following limit holds in Lp(ℝ2):
(4.10)Aψ(ξ/|ξ|)(ξ1/|ξ|)(φun1)+Aψ(ξ/|ξ|)(ξ2/|ξ|)(φun2)=Aψ(ξ/|ξ|)/|ξ|(∂1(φun1)+∂2(φun2))⟶0.
Multiplying (4.10) first by φvn1 and then by φvn2, integrating over ℝ2 and passing to the limit, we conclude from (2.2), due to the arbitrariness of ψ and φ:
(4.11)ξ1μ11+ξ2μ21=0,ξ1μ12+ξ2μ22=0.
Next, take(4.12)wnj=φAψ(ξ/|ξ|)/|ξ|(φunj)∈W1,p′(Rd),j=1,2.
From (4.9), we get
(4.13)〈(φvn1,-φvn2),∇wnj〉=-〈curl(φvn1,φvn2),wnj〉⟶0asn⟶∞,
for j=1,2. Rewriting it in the integral formulation, we obtain, from (2.2),
(4.14)ξ2μ11-ξ1μ12=0,ξ2μ21-ξ1μ22=0.
From the algebraic relations (4.11) and (4.14), we can easily conclude
(4.15)ξ1(μ11+μ22)=0,ξ2(μ11+μ22)=0,
implying that the distribution μ11+μ22 is supported on the set {ξ1=0}∩{ξ2=0}∩P=∅, which implies μ11+μ22≡0.
After insertingψ≡1 in the definition of H-distribution (2.2), we immediately reach the conclusion. This proof is similar to the L2 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(un) and (vn) converging weakly to zero in Lp(ℝd)d and Lp′(ℝd)d, and by μ denote d×d matrix H-distribution corresponding to some chosen subsequences of (un) and (vn).
Fromdivun→0 strongly in Wloc-1,p(ℝd), for φ1∈Cc(ℝd) and ψ∈Cκ(Sd-1), one has as in (4.10) that
(4.16)Aψ(ξ/|ξ|)(ξ/|ξ|)(φ1un)=Aψ(ξ/|ξ|)/|ξ|div(φ1un)⟶0stronglyinLp(Rd).
After forming a product with φ2vn, integrating and passing to the limit, we conclude that
(4.17)ξTμ=0,
namely, that the columns of μ are perpendicular to ξ.
On the other hand, fromcurlvn→0 strongly in Wloc-1,p′(ℝd)d×d, in an analogous way, we conclude that, for each row (denoted by μi) of μ, for all j,k, one has
(4.18)ξjμik-ξkμij=0,
so the rows of μ are proportional to ξ and μ=λ⊗ξ (a rank-one matrix), λi 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, trμ=λ·ξ=0, which implies the convergence un·vn⇀0, as in the two-dimensional situation.The above result is the well-known Murat’s div-curl lemma in the(Lp,Lp′)-setting [24, 25], which we state as a theorem.Theorem 4.3.
Let(un) and (vn) be vector-valued sequences converging to zero weakly in Lp(ℝd)d and Lp′(ℝd)d, respectively. Assume that the sequence (divun) is bounded in Lp(ℝd) and the sequence (curlvn) is bounded in Lp′(ℝd)d×d.
Then, the sequence(un·vn) converges to zero in the sense of distributions (or vaguely in the sense of Radon measures).



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*Source: 901084-2011-09-19.xml*