Radiance Scaling for Versatile Surface Enhancement

Keywords: Expressive rendering, NPR, Shape depiction

: Our novel Radiance Scaling technique enhances the depiction of surface shape under arbitrary illumination, with various materials, and in a wide range of rendering settings. In the left pair of images, we illustrate how surface features are enhanced mainly through enhancement of the specular shading term. Whereas on the right pair of images, we show the efficiency of our method on an approximation of a refractive material. Observe how various surface details are enhanced in both cases: around the eyes, inside the ear, and on the nose.

Introduction

The depiction of object shape has been a subject of increased interest in the Computer Graphics community since the work [START_REF] Saito | Comprehensible Ren-532 dering of 3-D Shapes[END_REF]. Inspired by their pioneering approach, many rendering techniques have focused on finding an appropriate set of lines to depict object shape. In contrast to line-based approaches, other techniques depict object shape through shading. Maybe the most widely used of these is Ambient Occlusion [START_REF] Pharr | GPU Gems[END_REF], which measures the occlusion of nearby geometry. Both types of techniques make drastic choices for the type of material, illumination and style used to depict an object: line-based approaches often ignore material and illumination and depict mainly sharp surface features, whereas occlusion-based techniques convey deep cavities for diffuse objects under ambient illumination. mination. However, this is not performed indiscriminately at every surface point and for the outgoing radiance only, but in a curvaturedependent manner and for each incoming light direction as in Light Warping. The main tool to achieve this enhancement is a novel scaling function presented in Section 4. In addition, Radiance Scaling takes material characteristics into account, which not only allows users to control accurately the enhancement per material component, but also makes the method easy to adapt to different rendering scenarios as shown in Section 5. Comparisons with related techniques and directions for future work are given in Section 6.

Previous work

Most of the work done for the depiction of shape in Computer Graphics concerns line-based rendering techniques. Since the seminal work of [START_REF] Saito | Comprehensible Ren-532 dering of 3-D Shapes[END_REF], many novel methods (e.g., [Nienhaus and Döllner 2004;[START_REF] Ohtake | Ridge-514 valley lines on meshes via implicit surface fitting[END_REF][START_REF] Decarlo | Suggestive Contours for Conveying Shape[END_REF][START_REF] Judd | Apparent Ridges for Line Drawing[END_REF][START_REF] Lee | 504 Line drawings via abstracted shading[END_REF][START_REF] Goodwin | Isophote distance: a shading approach to artistic stroke thickness[END_REF][START_REF] Kolomenkin | Demarcating Curves for Shape Illustration[END_REF][START_REF] Zhang | Laplacian 550 Lines for Real Time Shape Illustration[END_REF]) have been proposed.

Most of these techniques focus on depicting shape features directly, and thus make relatively little use of material or illumination information, with the notable exception of [START_REF] Lee | 504 Line drawings via abstracted shading[END_REF].

A number of shading-based approaches have also shown interesting abilities for shape depiction. The most widely used of these techniques is Ambient Occlusion [START_REF] Pharr | GPU Gems[END_REF], which measures the occlusion of nearby geometry. The method rather tends to depict deep cavities, whereas shallow (yet salient) surface details are often missed or even smoothed out. Moreover, enhancement only occurs implicitly (there is no direct control over the shading features to depict), and the method is limited to diffuse materials and ambient lighting. It is also related to Accessibility shading techniques (e.g., [START_REF] Miller | Efficient Algorithms for Local and Global 507 Accessibility Shading[END_REF]]), which conveys information about concavities of a 3D object. Ritshel et al. [2008] addresses limitations on the type of material or illumination. It consists in applying the Cornsweet Illusion effect to outgoing radiance on an object surface. The approach provides interesting enhancement not only with diffuse materials, but also with glossy objects, shadows and textures. However, the method is applied indiscriminately to all these effects, and thus enhances surface features only implicitly, when radiance happens to be correlated with surface shape. Moreover, it produces artifacts when applied to glossy objects: material appearance is then strongly altered and objects tend to look sharper than they really are. Hence, the method is likely to create noticeable artifacts when applied to highly reflective or refractive materials as well.

The recent 3D Unsharp Masking technique of

In this paper, we rather seek a technique that enhances object shape explicitly, with intuitive controls for the user. Previous methods [START_REF] Kindlmann | Curvature-Based Transfer Functions for Direct Volume Rendering: Methods and Applications[END_REF][START_REF] Cignoni | A simple Normal Enhancement technique for Interactive Nonphotorealistic Renderings[END_REF][START_REF] Rusinkiewicz | Ex-529 aggerated Shading for Depicting Shape and Detail[END_REF][START_REF] Vergne | 542 Apparent relief: a shape descriptor for stylized shading[END_REF]Vergne et al. 2009] differ in the geometric features they enhance and on the constraints they put on materials, illumination or style. For instance, Exaggerated Shading [START_REF] Rusinkiewicz | Ex-529 aggerated Shading for Depicting Shape and Detail[END_REF]] makes use of normals at multiple scales to define surface relief and relies on a Half-Lambertian to depict relief at grazing angles. The most recent and general of these techniques is Light Warping [Vergne et al. 2009]. It makes use of a view-centered curvature tensor to define surface features, which are then enhanced by locally stretching or compressing reflected light patterns around the view direction. Although this technique puts no constraint on the choice of material or illumination, its effectiveness decreases with lighting environments that do not exhibit natural statistics. It also requires a dense sampling of illumination, and is thus not adapted to simplified lighting such as found in video games, or to the use of precomputed radiance methods. Moreover, highly reflective or refractive materials produce complex warped 130 patterns that tend to make rendering less legible. The key observation of this paper is that explicitly correlating re-

L ′ (p → e) = Z Ω ρ(e, ℓ ℓ ℓ)(n • ℓ ℓ ℓ) σ(p, e, ℓ ℓ ℓ) L(p ← ℓ ℓ ℓ) dℓ ℓ ℓ (1)
where L ′ is the enhanced radiance, p is a surface point, e is the di-149 rection toward the eye, n is the surface normal at p, Ω is the hemi-150 sphere of directions around n, ℓ ℓ ℓ is a light direction, ρ is the material 151 BRDF, σ is a scaling function and L is the incoming radiance.

152

The scaling function is a short notation for σα,γ(κ(p), δ(e, ℓ ℓ ℓ)).

σα,γ(κ, δ) = αγe κ + δ(1 -α(1 + γe κ )) (α + δ(γe κ -α(1 + γe κ ))) (2)
where α ∈ (0, 1) controls the location of the scaling-invariant point The exact same reflectance mapping functions could be used with more complex precomputed radiance transfer methods.

Perfectly diffuse materials

For diffuse materials, Ambient Occlusion [START_REF] Pharr | GPU Gems[END_REF] and Prefiltered Environment Maps [START_REF] Kautz | Unified Approach to Prefiltered Environment Maps[END_REF]] are among the most widely used techniques to precompute radiance data. We show in the following a similar approximation used in conjunction with Radiance Scaling. The BRDF is first considered constant diffuse: ρ(e, ℓ ℓ ℓ) = ρ d . We then consider only direct illumination from an environment map: L(p ← ℓ ℓ ℓ) = V (ℓ ℓ ℓ)Lenv(ℓ ℓ ℓ) where V ∈ {0, 1} is a visibility term and Lenv is the environment map.

Equation 3 then becomes:

L ′ (p → e) = σ(p, e) ρ d Z Ω (n • ℓ ℓ ℓ) V (ℓ ℓ ℓ) Lenv(ℓ ℓ ℓ) dℓ ℓ ℓ
We then approximate the enhanced radiance with

L ′ (p → e) ≃ σ(p, e) ρ d A(p) L(n)
with A(p) the ambient occlusion stored at each vertex, and L an irradiance average stored in a prefiltered environment map:

A(p) = R Ω (n • ℓ ℓ ℓ)V (ℓ ℓ ℓ)dℓ ℓ ℓ , L(n) = R Ω Lenv(ℓ ℓ ℓ)dℓ ℓ ℓ
For perfectly diffuse materials, we use the reflectance mapping The video also shows results when the method is applied to an object with a mirror-like material.

Discussion

We first compare Radiance Scaling with previous methods in Section 6.1, with a focus on Light Warping [Vergne et al. 2009] since it relies on the same surface features. We then discuss limitations and avenues for future work in Section 6.2.

Comparisons with previous work

Our approach is designed to depict local surface features, and is difficult to compare with approaches such as Accessibility Shading that consider more of the surrounding geometry. Accessibility Shading characterizes how easily a surface may be touched by a spherical probe, and thus tends to depict more volumetric features.

However, for surfaces where small-scale relief dominates largescale variations (such as carved stones or roughly textured statues), the spherical probe acts as a curvature measure. In this case, Accessibility Shading becomes similar to Mean Curvature Shading, which is a special case of Radiance Scaling as seen in Figure 4-b.

A technique related to Accessibility Shading is Ambient Occlusion: indeed, measuring occlusion from visible geometry around a surface point is another way of probing a surface. Ambient Occlusion is more efficient at depicting proximity relations between objects (such as contacts), and deep cavities. However, as seen in Figure 6 patterns humans are able to make use for perceiving shape is obvi-442 ously much more diverse than simple alternations of bright and dark 443 colors in convexities and concavities [START_REF] Koenderink | The visual neurosciences[END_REF]]. And 444 these patterns are likely to be dependent on the main illumination 445 direction(e.g., [START_REF] Ho | How direction of illumination affects visually perceived surface roughness[END_REF][START_REF] Caniard | Distortion in 3D shape estimation with changes in illumination[END_REF]O'Shea 446 et al. 2008]), material characteristics (e.g., [START_REF] Adelson | On seeing stuff: the perception of materials by humans and machines[END_REF]Vangorp 447 et al. 2007]), motion (e.g., [START_REF] Pont | Computer Analysis of Images and Patterns[END_REF][START_REF] Adato | Toward a theory of shape from specular flow[END_REF]), and 448 silhouette shape (e.g., [START_REF] Fleming | Specular reflections and the perception of shape[END_REF]). Characterizing such 449

33

  More versatile shape enhancement techniques are required to ac-34 commodate the needs of modern Computer Graphics applications. 35 They should work with realistic as well as stylized rendering to 36 adapt to the look-and-feel of a particular movie or video game pro-37 duction. A wide variety of materials should be taken into account, 38 such as diffuse, glossy and transparent materials, with specific con-39 trols for each material component. A satisfying method should 40 work for various illumination settings ranging from complex illumi-41 nation for movie production, to simple or even precomputed illumi-42 nation for video games. On top of these requirements, enhancement 43 methods should be fast enough to be incorporated in interactive ap-44 plications or to provide instant feedback for previewing.

133

  flected lighting variations to surface feature variations leads to an 134 improved depiction of object shape. For example, consider a high-135 light reflected off a glossy object; by increasing reflected light in-136 tensity in convex regions and decreasing it in concave ones, the 137 highlight looks as if it is attracted toward convexities and repelled 138 from concavities (see Figure 1-left). Such an adjustment improves 139 the distinction between concave and convex surface features, and 140 does not only take surface features into account, but also material 141 characteristics. Indeed, reflected light intensity has an altogether 142 different distribution across the surface depending on whether the 143 material is glossy or diffuse for instance. 144 The main idea of Radiance Scaling is thus to adjust reflected light 145 intensity per incoming light direction in a way that depends on both 146 surface curvature and material characteristics. Formally, we rewrite 147 the reflected radiance equation as follows:

  148

153

  The curvature mapping function κ(p) : R 3 → [-1, 1] computes 154 normalized curvature values, where -1 corresponds to maximum 155 concavities, 0 to planar regions and 1 to maximum convexities. We 156 call δ(e, ℓ ℓ ℓ) : Ω 2 → [0, 1] the reflectance mapping function. It 157 computes normalized values, where 0 corresponds to minimum re-158 flected intensity, and 1 to maximum reflected intensity. Intuitivelly, 159 it helps identify the light direction that contributes the most to re-160 flected light intensity. 161 We describe the formula for the scaling function and the choice 162 of curvature mapping function in Section 4. We then show how 163 Radiance Scaling is easily adapted to various BRDF and illumina-164 tion scenarios by a proper choice of reflectance mapping function in Equation 1 is a function of two variables: a nor-168 malized curvature and a normalized reflectance. Both variables are 169 themselves functions, but for clarity of notation, we use κ and δ in 170 the following. We require the scaling function to be monotonic, so 171 that no new shading extremum is created after scaling. Another re-172 quirement is that for planar surface regions, the function must have 173 no influence on reflected lighting. The following function fulfills 174 these requirements (see Figure 2):

  175

176

  of σ and γ ∈ [0, ∞) is the scaling magnitude. The scaling-invariant 177 point is a handy parameter to control how variations in shading de-178 pict surface feature variations. For convex features, reflected light-179 ing intensities above α are brightened and those below α are dark-180 ened. For concave features, the opposite effect is obtained. This is 181 illustrated in Figure 3.

  182

Figure 2 :

 2 Figure 2: Two plots of a set of scaling functions with different scaling-invariant points (left: α = 0.2; right:α = 0.8), and using increasing curvatures κ = {-1,-1/2, 0, 1/2,1 }.

Figure 3 :Figure 4 264

 34 Figure 3: The effect of scaling parameters. Left: no scaling (γ = 0). Middle: scaling with a low scaling-invariant point (α = 0.2): convexities are mostly brightened. Right: scaling with a high scaling-invariant point (α = 0.8): convexities are brightened in the direction of the light source, but darkened away from it.

Figure 4 :Figure 5 Figure 5 :

 455 Figure 4: Radiance Scaling in simple lighting scenarios: (a) Each lobe of Phong's shading model is scaled independently to reveal shape features such as details in the hair. (b) Radiance Scaling is equivalent to Mean Curvature Shading when applied to an ambient lobe (we combine it with diffuse shading in this Figure). (c) Surface features are also convincingly enhanced with Cartoon Shading, as with this little girl character (e.g., observe the right leg and foot, the ear, the bunches, or the region around the nose). (d) Radiance Scaling is efficient even with sub-Lambertian materials, as in this example of a moon modeled by a sphere and a detailed normal map.

333Figure 6 Figure 6 :

 66 Figure 6-a shows the warping of prefiltered environment maps us-340

Figure 1 -

 1 Figure 1-right shows how Radiance Scaling enhances surface features with a simple approximation of a purely refractive material.

Figure 7 :

 7 Figure 7: This plot gives the performances obtained with the scene shown in Figure 5 without enhancement, and with both Radiance Scaling and Light Warping. The 3D model is composed of 1, 652, 528 polygons. While the time for rendering a single frame increases linearly with the number of light samples in all cases, our novel method is linearly faster than Light Warping.

Figure 8 :

 8 Comparison with Light Warping. Top row: Light warping image. Bottom row: Radiance Scaling. (a-b) Both methods show similar enhancement abilities when used with a diffuse material and a natural illumination environment: convexities exhibit brighter colors, and concavities darker colors in most cases. For some orientation of the viewpoint relative to the environment, Light Warping may reverse this effect though (concavities are brighter, convexities darker) while Radiance Scaling does not. (c-d) The methods are most different with shiny objects, shown with two illumination orientations as well. Light Warping, as seen in Figure 7. However, the two methods are not qualitatively equivalent, as shown in Figure 8. For dif-414 fuse materials and with natural illumination, the two methods pro-415 duce similar results: concavities are depicted with darker colors, 416 and convexities with brighter colors. However, for some orien-417 tations of the viewpoint relative to the environment illumination, 418 Light Warping may reverse this effect, since rays are attracted to-419 ward or away from the camera regardless of light source locations. 420 Radiance Scaling does not reverse tone in this manner. The main 421 difference between the two techniques appears with shiny materi-422 als. In this case, the effect of enhancement on illumination is more 423 clearly visible: Light Warping modulates lighting frequency, while 424 Radiance Scaling modulates lighting intensity, as is best seen in the 425 supplementary video.

4266. 2

 2 Directions for future work 427 We have shown that the adjustment of reflected light intensities, 428 a process we call Radiance Scaling, provides a versatile approach 429 to the enhancement of surface shape through shading. However, 430 when the enhancement magnitude is pushed to extreme values, our 431 method alters material appearance. This is because variations in 432 shape tend to dominate variations due to shading. An exciting av-433 enue of future work would be to characterize perceptual cues to 434 material appearance and preserve them through enhancement. 435 Although Radiance Scaling produces convincing enhancement in 436 many rendering scenarios, there is still room for alternative en-437 hancement techniques. Indeed, our approach makes two assump-438 tions that could be dropped in future work: 1) concave and convex 439 features have inverse effects on scaling; and 2) enhancement is ob-440 tained by local differential operators. The class of reflected lighting

  441


E Yu Panov

Ultraparabolic H-measures and compensated compactness

research documents, whether they are published or not. The documents may come

Ultraparabolic H-measures and compensated compactness

E.

Introduction

Recall the classical results of the compensated compactness theory ( see [START_REF] Murat | Compacité par compensation[END_REF][START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] ).

Suppose that Ω is an open subset of R n , and a sequence u r = (u 1r (x), . . . , u N r (x)) ∈ L 2 (Ω, R N ), r ∈ N, weakly converges to a vectorfunction u(x) in L 2 (Ω, R N ). Assume that a sαk are real constants for s = 1, . . . , m, α = 1, . . . , N, k = 1, . . . , n, and the sequences of distributions

are strongly precompact in the space H -1 loc (Ω) . = W -1 2,loc (Ω). Hereafter, we denote by W -1 p,loc (Ω), 1 ≤ p ≤ ∞ the locally convex space consisting of distributions v ∈ D ′ (Ω) such that the distribution f v belongs to the Sobolev space W -1 p . = W -1 p (R n ) for all f (x) ∈ C ∞ 0 (Ω). The topology in W -1 p,loc (Ω) is generated by the family of semi-norms u → uf W -1 p , f (x) ∈ C ∞ 0 (Ω). Introduce the set

Λ = λ ∈ R N | ∃ξ ∈ R n , ξ = 0 : N α=1 n k=1
a sαk λ α ξ k = 0 ∀s = 1, . . . , m . Now, let q(u) = N α,β=1 q αβ u α u β be a quadratic functional on R l such that q(λ) ≥ 0 for all λ ∈ Λ, and q(u r ) → v weakly in the sense of distributions on Ω ( in D ′ (Ω) ).

Then, under the above assumptions,

q(u(x)) ≤ v in D ′ (Ω)
(the weak low semicontinuity). In particular, if q(λ) = 0 on Λ then v = q(u).

1

In this paper we generalize this result to the case when the differential constraints may contain second order terms, while all the coefficients are variable and may be discontinuous. Thus, assume that a sequence u r (x) is bounded in L p loc (Ω, R N ), 2 ≤ p ≤ ∞ and converges weakly in D ′ (Ω) to a vector-function u(x) as r → ∞. Let d = p/(p -1) if p < ∞, and d > 1 if p = ∞. Assume that the sequences

N α=1 n k=1 ∂ x k (a sαk u αr ) + N α=1 n k,l=ν+1 ∂ x k x l (b sαkl u αr ), s = 1, . . . , m (2)
are pre-compact in the anisotropic Sobolev space W -1,-2 d,loc (Ω), which will be defined later in Section 2. Here ν is an integer number between 0 and n, and the coefficients a sαk = a sαk (x), b sαkl = b sαkl (x) belong to the space L 2q loc (Ω), q = p/(p -2) ( q = 1 in the case p = ∞ ), if p > 2, and to the space C(Ω) if p = 2. One example is given by p = ∞, q = 1 and corresponds to the case when the functions u r (x) are uniformly locally bounded.

We introduce the set Λ ( here i = √ -1 ):

Λ = Λ(x) = λ ∈ C N | ∃ξ ∈ R n , ξ = 0 : N α=1 i ν k=1 a sαk (x)ξ k - n k,l=ν+1
b sαkl (x)ξ k ξ l λ α = 0 ∀s = 1, . . . , m .

Consider the quadratic form q(x, u) = Q(x)u • u, where Q(x) is a symmetric matrix with coefficients q αβ (x), α, β = 1, . . . , N and u • v denotes the scalar multiplication on R N . The form q(x, u) can be extended as Hermitian form on C N by the standard relation

q(x, u) = N α,β=1 q αβ (x)u α u β ,
where we denote by u the complex conjugation of u ∈ C. We suppose that the coefficients q αβ (x) ∈ L q loc (Ω) if p > 2, and q αβ (x) ∈ C(Ω) if p = 2. Now, let the sequence q(x, u r ) → v as r → ∞ weakly in D ′ (Ω). Since for each α, β = 1, . . . , N the sequences u αr (x)u βr (x) are bounded in L p/2 loc (Ω) (here p/2 = ∞ for p = ∞) then, passing to a subsequence if necessary, we may claim that u αr (x)u βr (x)

→ r→∞ ζ αβ (x)
weakly in L p/2 loc (Ω) if p > 2 (hereafter, the weak convergence in L ∞ loc (Ω) is understood in the sense of the weak- * topology), and weakly in the space M loc (Ω) of locally finite measures on Ω if p = 2. In view of the relation 1 q + 2 p = 1 this implies that

q(x, u r ) → r→∞ N α,β=1 q αβ (x)ζ αβ (x) weakly in M loc (Ω) (weakly in L 1 loc (Ω) if p > 2) and therefore v(x) = N α,β=1 q αβ (x)ζ αβ (x).
In particular, v = v(x) ∈ L 1 loc (Ω) for p > 2 and v ∈ M loc (Ω) for p = 2. Our main result is the following Theorem 1. Assume that q(x, λ) ≥ 0 for all λ ∈ Λ(x), x ∈ Ω. Then q(x, u(x)) ≤ v ( in the sense of measures ).

Main concepts

To prove Theorem 1 we will use the techniques of H-measures. Let

F (u)(ξ) = R n e -2πiξ•x u(x)dx, ξ ∈ R n ,
be the Fourier transformation extended as a unitary operator on the space u

(x) ∈ L 2 (R n ), let S = S n-1 = { ξ ∈ R | |ξ| = 1 } be the unit sphere in R n .
The concept of an H-measure corresponding to some sequence of vector-valued functions bounded in L 2 (Ω) was introduced by Tartar [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF] and Gerárd [START_REF] Gerárd | Microlocal defect measures[END_REF] on the basis of the following result. For r ∈ N let U r (x) = U 1 r (x), . . . , U N r (x) ∈ L 2 (Ω, R N ) be a sequence weakly convergent to the zero vector.

Proposition 1 (see [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF]Theorem 1.1]). There exists a family of complex Borel measures µ = µ αβ N α,β=1 in Ω×S and a subsequence of U r (x) (still denoted U r ) such that

µ αβ , Φ 1 (x)Φ 2 (x)ψ(ξ) = lim r→∞ R n F (U α r Φ 1 )(ξ)F (U β r Φ 2 )(ξ)ψ ξ |ξ| dξ (4)
for all Φ 1 (x), Φ 2 (x) ∈ C 0 (Ω) and ψ(ξ) ∈ C(S).

The family µ = µ αβ N α,β=1 is called the H-measure corresponding to U r (x). In [START_REF] Antonić | H-measures and variants applied to parabolic equations[END_REF] the new concept of parabolic H-measures was introduced. Here we need the more general variant of this concept recently developed in [START_REF] Panov | Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property[END_REF]. Suppose that X ⊂ R n is a linear subspace, X ⊥ is its orthogonal complement, P 1 , P 2 are orthogonal projections on X, X ⊥ , respectively. We denote for ξ ∈ R n ξ = P 1 ξ,

ξ = P 2 ξ, so that ξ ∈ X, ξ ∈ X ⊥ , ξ = ξ + ξ. Let S X = { ξ ∈ R n | | ξ| 2 + | ξ| 4 = 1 }.
Then S X is a compact smooth manifold of codimension 1; in the case when X = {0} or X = R n , it coincides with the unit sphere

S = {ξ ∈ R n | |ξ| = 1 }. Let us define a projection π X : R n \ {0} → S X by π X (ξ) = ξ (| ξ| 2 + | ξ| 4 ) 1/2 + ξ (| ξ| 2 + | ξ| 4 ) 1/4 .
Remark that in the case when X = {0} or X = R n , π X (ξ) = ξ/|ξ| is the orthogonal projection on the sphere. We denote p(ξ) = (| ξ| 2 + | ξ| 4 ) 1/4 . The following useful property of the projection π X holds (see [START_REF] Panov | Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property[END_REF]Lemma 1]).

Lemma 1. Let ξ, η ∈ R n , max(p(ξ), p(η)) ≥ 1. Then |π X (ξ) -π X (η)| ≤ 6|ξ -η| max(p(ξ), p(η))
.

Proof. For ξ ∈ R n , α > 0, we define ξ α = α 2 ξ + α ξ. Observe that for all α > 0, π X (ξ α ) = π X (ξ). Without loss of generality we may suppose that p(ξ) ≥ p(η), and in particular p(ξ

) ≥ 1. Remark that π X (ξ) = ξ α , π X (η) = η β , where α = 1/p(ξ), β = 1/p(η). Therefore, |π X (ξ) -π X (η)| = |ξ α -η β | ≤ |ξ α -η α | + |η α -η β | = α 4 | ξ -η| 2 + α 2 | ξ -η| 2 1/2 + (β 2 -α 2 ) 2 |η| 2 + (β -α) 2 |η| 2 1/2 ≤ α|ξ -η| + (β -α) (β + α) 2 |η| 2 + |η| 2 1/2 . (5)
Here we take into account that α ≤ 1 and therefore α 4 ≤ α 2 . Since

(β + α) 2 ≤ 4β 2 = 4(|η| 2 + |η| 4 ) -1/2 ≤ 4/|η|,
we have the estimate

(β + α) 2 |η| 2 + |η| 2 ≤ 4(|η| + |η| 2 ) ≤ 4 2(|η| 2 + |η| 4 ) 1/2 ≤ 6(p(η)) 2 . ( 6
)
Concerning the term β -α, we estimate it as follows

β -α = p(ξ) -p(η) p(ξ)p(η) = p(ξ) 4 -p(η) 4 p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) = | ξ| 2 -|η| 2 + | ξ| 4 -|η| 4 p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) ≤ (| ξ| + |η|)| ξ -η| + (| ξ| + |η|)(| ξ| 2 + |η| 2 )| ξ -η| p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) ≤ | ξ| + |η| + (| ξ| + |η|)(| ξ| 2 + |η| 2 ) p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) |ξ -η| ≤ (p(ξ)) 2 + (p(η)) 2 + (p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) |ξ -η| ≤ 1 + p(ξ) + p(η) p(ξ) + p(η) |ξ -η| p(ξ)p(η) ≤ 2|ξ -η| p(ξ)p(η) . (7)
Here we use that ξ ≤ (p(ξ)) 2 , ξ ≤ p(ξ), η ≤ (p(η)) 2 , η ≤ p(η), and that p(ξ) + p(η) ≥ 1. Now it follows from ( 5), ( 6), [START_REF] Pedregal | Parametrized measures and variational principles[END_REF] that

|π X (ξ) -π X (η)| ≤ |ξ -η| p(ξ) + 2 √ 6|ξ -η| p(ξ) ≤ 6|ξ -η| p(ξ) = 6|ξ -η| max(p(ξ), p(η)) ,
as was to be proved.

Let b(x) ∈ C 0 (R n ), a(z) ∈ C(S X
). We introduce the pseudo-differential operators B, A with symbols b(x), a(π X (ξ)), respectively. These operators are multiplication operators Bu(x) = b(x)u(x), F (Au)(ξ) = a(π X (ξ))F (u)(ξ). Obviously, the operators B, A are well-defined and bounded in L 2 . As was proved in [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF], in the case when S X = S, π X (ξ) = ξ/|ξ| the commutator [A, B] = AB -BA is a compact operator. In [START_REF] Panov | Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property[END_REF], using the assertion of Lemma 1, we extend this result for the general case ( in the case dim X = 1 this was done in [START_REF] Antonić | H-measures and variants applied to parabolic equations[END_REF] ). For completeness we give the details below.

Lemma 2. The operator

[A, B] is compact in L 2 . Proof. We can find sequences a k (z) ∈ C ∞ (S X ), b k (x) ∈ C ∞ (R n ), k ∈ N with the following properties: F (b k )(ξ) ∈ C ∞ 0 (R n ), and a k (z) → a(z), b k (x) → b(x)
as k → ∞ uniformly on S X , R n , respectively. Then the sequences of the operators A k , B k with symbols a k (π X (ξ)), b k (x) converge as k → ∞ to the operators A, B, respectively (in the operator norm). Therefore,

[A k , B k ] → k→∞ [A, B] and it is sufficient to prove that the operators [A k , B k ] are compact for all k ∈ N ( then [A, B] is a compact operator as a limit of compact operators ). Let u = u(x) ∈ L 2 (R n ). Then by the known property F (bu)(ξ) = F (b) * F (u)(ξ) = F (b)(ξ -η)F (u)(η)dη, F ([A k , B k ]u)(ξ) = F (A k B k u)(ξ) -F (B k A k u)(ξ) = a k (π X (ξ))F (b k u)(ξ) -F (b k A k u)(ξ) = R n (a k (π X (ξ)) -a k (π X (η)))F (b k )(ξ -η)F (u)(η)dη.
We have to prove that the integral operator

Kv(ξ) = R n k(ξ, η)v(η)dη with the kernel k(ξ, η) = (a k (π X (ξ)) -a k (π X (η)))F (b k )(ξ -η) is compact on L 2 (R n ). Since a k ∈ C ∞ (S X ) then by Lemma 1 |a k (π X (ξ)) -a k (π X (η))| ≤ C |ξ -η| max(p(ξ), p(η)) for max(p(ξ), p(η)) ≥ 1, where C = const. Thus for all ξ, η ∈ R n such that max(p(ξ), p(η)) > m > 1 |a k (π X (ξ)) -a k (π X (η))| ≤ C m |ξ -η|. (8)
Let χ m (ξ, η) be the indicator function of the set

{ (ξ, η) ∈ R 2n | max(p(ξ), p(η)) ≤ m }, and
k m (ξ, η) = χ m (ξ, η)(a k (π X (ξ)) -a k (π X (η)))F (b k )(ξ -η), r m (ξ, η) = (1 -χ m (ξ, η))(a k (π X (ξ)) -a k (π X (η)))F (b k )(ξ -η). Then k(ξ, η) = k m (ξ, η) + r m (ξ, η) and K = K m + R m ,
where K m , R m are integral operators with the kernels k m (ξ, η), r m (ξ, η), respectively. Since the function k m (ξ, η) is bounded and compactly supported then the operator K m is a Hilbert-Schmidt operator, which is compact. On the other hand, in view of ( 8)

|R m v(ξ)| ≤ C m R n |(ξ -η)F (b k )(ξ -η)||v(η)|dη = [|ξF (b k )| * |v|](ξ)
and, by the Young inequality, for every

v ∈ L 2 (R n ) R m v 2 ≤ C m ξF (b k ) 1 v 2 .
Therefore, R m ≤ const/m and R m → 0 as m → ∞. We conclude that K m → K and therefore K is a compact operator, as a limit of compact operators. This completes the proof.

The ultra-parabolic H-measure µ αβ , α, β = 1, . . . , N corresponding to a subspace X ⊂ R n and a sequence U r (x) ∈ L 2 (Ω, R N ), weakly convergent to the zero vector, is defined on Ω×S X by the relation similar to (4):

∀Φ 1 (x), Φ 2 (x) ∈ C 0 (Ω), ψ(ξ) ∈ C(S X ) µ αβ , Φ 1 (x)Φ 2 (x)ψ(ξ) = lim r→∞ R n F (Φ 1 U α r )(ξ)F (Φ 2 U β r )(ξ)ψ(π X (ξ))dξ. ( 9
)
The existence of the H-measure µ αβ is proved exactly in the same way as in [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF], using the statement of Lemma 2. For completeness we give the details below.

Proposition 2. There exist a family of complex Borel measures µ = µ αβ N α,β=1 in Ω × S X and a subsequence of U r (x) (still denoted by U r ) such that relation [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] holds for all Φ 1 (x), Φ 2 (x) ∈ C 0 (Ω), ψ(ξ) ∈ C(S X ). Besides, the matrix-valued measure µ is Hermitian and positive definite, that is, for each

ζ = (ζ 1 , . . . , ζ N ) ∈ C n the measure µζ • ζ = N α,β=1 µ αβ ζ α ζ β ≥ 0. Proof. Denote I αβ r (Φ 1 , Φ 2 , ψ) = R n F (Φ 1 U α r )(ξ)F (Φ 2 U β r )(ξ)ψ(π X (ξ))dξ
and observe that, by the Buniakovskii inequality and the Plancherel identity,

|I αβ r | ≤ Φ 1 ∞ Φ 2 ∞ ψ ∞ • U α r L 2 (K) U β r L 2 (K) ,
where K ⊂ Ω is a compact containing supports of Φ 1 and Φ 2 . In view of the weak convergence of sequences U α r in L 2 (K) these sequences are bounded in L 2 (K). Therefore, for some constant

C K we have U α r 2 L 2 (K) ≤ C K for all r ∈ N, α = 1, . . . , N. Hence, |I αβ r (Φ 1 , Φ 2 , ψ)| ≤ C K Φ 1 ∞ Φ 2 ∞ ψ ∞ (10)
and the sequences I αβ r are bounded. Let D be a countable dense set in (C 0 (Ω)) 2 × C(S X ). Using the standard diagonal process, we can extract a subsequence U r (we keep the notation U r for this subsequence) such that

I αβ r (Φ 1 , Φ 2 , ψ) → r→∞ I αβ (Φ 1 , Φ 2 , ψ) (11)
for all triples (Φ 1 , Φ 2 , ψ) ∈ D. By estimate [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF] we see that sequences

I αβ r (Φ 1 , Φ 2 , ψ) are uniformly continuous with respect to (Φ 1 , Φ 2 , ψ) ∈ (C 0 (Ω)) 2 × C(S X ) and since D is dense in (C 0 (Ω)) 2 × C(S X ), we conclude that limit relation (11) holds for all Φ 1 (x), Φ 2 (x) ∈ C 0 (Ω), ψ(ξ) ∈ C(S X ).
Passing in [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF] to the limit as r → ∞, we derive that for all Φ

1 (x), Φ 2 (x) ∈ C 0 (Ω), ψ(ξ) ∈ C(S X ) |I αβ (Φ 1 , Φ 2 , ψ)| ≤ C K Φ 1 ∞ Φ 2 ∞ ψ ∞ , (12)
with K = supp Φ 1 ∪ supp Φ 2 . Now, we observe that

I αβ r (Φ 1 , Φ 2 , ψ) = (Φ 1 U α r , A(Φ 2 U β r )) 2 , ( 13
)
where A is a pseudo-differential operator on L 2 = L 2 (R n ) with symbol ψ(π X (ξ)), and (•, •) 2 is the scalar product in L 2 . Let B be a pseudo-differential operator on L 2 with symbol Φ 2 (x), and let ω(x) ∈ C 0 (R n ) be a function such that ω(x) ≡ 1 on supp Φ 2 . Then

A(Φ 2 U β r ) = AB(ωU β r ) = BA(ωU β r ) + [A, B](ωU β r ). ( 14
) Since ωU β r → 0 as r → ∞ weakly in L 2 while, by Lemma 2, the operator [A, B] is compact on L 2 , we claim that [A, B](ωU β r ) → 0 as r → ∞ strongly in L 2 . Since the sequence Φ 1 U α r is bounded in L 2 , we conclude that (Φ 1 U α r , [A, B](ωU β r
)) 2 → 0 as r → ∞. It follows from this limit relation and ( 13), ( 14) that

lim r→∞ (Φ 1 U α r , BA(ωU β r )) 2 = lim r→∞ I αβ r (Φ 1 , Φ 2 , ψ) = I αβ (Φ 1 , Φ 2 , ψ).
Taking into account that

(Φ 1 U α r , BA(ωU β r )) 2 = R n Φ 1 (x)Φ 2 (x)U α r (x)A(ωU β r )(x)dx,
we find that

I αβ (Φ 1 , Φ 2 , ψ) = Ĩαβ (Φ 1 Φ 2 , ψ),
where Ĩαβ (Φ, ψ) is a bilinear functional on C 0 (Ω) × C(S X ) for each α, β = 1, . . . , N. Taking in the above relation

Φ 1 = Φ(x)/ |Φ(x)| (we set Φ 1 (x) = 0 if Φ(x) = 0), Φ 2 = |Φ(x)|
, where Φ(x) ∈ C 0 (Ω), we find with the help of ( 12) that

| Ĩαβ (Φ, ψ)| = |I αβ (Φ 1 , Φ 2 , ψ)| ≤ C K Φ 1 ∞ Φ 2 ∞ ψ ∞ = C K Φ ∞ ψ ∞ , K = supp Φ.
This estimate shows that the functionals Ĩαβ (Φ, ψ) are continuous on C 0 (Ω) × C(S X ). Now, we observe that for nonnegative Φ(x) and ψ(ξ) the matrix Ĩ . = { Ĩαβ (Φ, ψ)} N α,β=1 is Hermitian and positive definite. Indeed, taking Φ

1 (x) = Φ 2 (x) = Φ(x), we find Ĩαβ (Φ, ψ) = I αβ (Φ 1 , Φ 1 , ψ) = lim r→∞ R n F (Φ 1 U α r )(ξ)F (Φ 1 U β r )(ξ)ψ(π X (ξ))dξ. ( 15
) For ζ = (ζ 1 , . . . , ζ N ) ∈ C N we have, in view of (15), Ĩζ • ζ = N α,β=1 Ĩαβ (Φ, ψ)ζ α ζ β = lim r→∞ R n |F (Φ 1 V r )(ξ)| 2 ψ(π X (ξ))dξ ≥ 0, where V r (x) = N α=1 U α r ζ α .
The above relation proves that the matrix Ĩ is Hermitian and positive definite.

We see that for any

ζ ∈ C n the bilinear functional Ĩ(Φ, ψ)ζ • ζ is continuous on C 0 (Ω) × C(S X ) and nonnegative, that is, Ĩ(Φ, ψ)ζ • ζ ≥ 0 whenever Φ(x) ≥ 0, ψ(ξ) ≥ 0.
It is rather well known ( see for example [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF]Lemma 1.10] ), that such a functional is represented by integration over some unique locally finite non-negative Borel measure

µ = µ ζ (x, ξ) ∈ M loc (Ω × S X ): Ĩ(Φ, ψ)ζ • ζ = Ω×S X Φ(x)ψ(ξ)dµ ζ (x, ξ).
As a function of the vector ζ, µ ζ is a measure valued Hermitian form. Therefore,

µ ζ = N α,β=1 µ αβ ζ α ζ β (16)
with measure valued coefficients µ αβ ∈ M loc (Ω × S X ), which can be expressed as follows

µ αβ = [µ eα+e β + iµ eα+ie β ]/2 -(1 + i)(µ eα + µ e β )/2,
where e 1 , . . . , e N is the standard basis in C N , and i 2 = -1. By ( 16)

Ĩ(Φ, ψ)ζ • ζ = l α,β=1 µ αβ , Φ(x)ψ(ξ) ζ α ζ β and since Ĩ(Φ, ψ)ζ • ζ = l α,β=1 Ĩαβ (Φ, ψ)ζ α ζ β ,
then, comparing the coefficients, we find that

µ αβ , Φ(x)ψ(ξ) = Ĩαβ (Φ, ψ). (17)
In particular,

µ αβ , Φ 1 (x)Φ 2 (x)ψ(ξ) = I αβ (Φ 1 , Φ 2 , ψ) = lim r→∞ R n F (Φ 1 U α r )(ξ)F (Φ 2 U β r )(ξ)ψ(π X (ξ))dξ.
To complete the proof, observe that for each ζ ∈ C N the measure N α,β=1

µ αβ ζ α ζ β = µ ζ ≥ 0.
Hence, µ is Hermitian and positive definite.

As it follows from the above Proposition, the matrix with component µ αβ , g(x, ξ) is Hermitian and positive definite for each real nonnegative g(x, ξ) ∈ C 0 (Ω × S X ).

Remark 1. We can replace the function ψ(π X (ξ)) in relation ( 9) by a function ψ(ξ

) ∈ C(R n ), which equals ψ(π X (ξ)) for large |ξ|. Indeed, since Φ 2 (x) is a function with compact support, Φ 2 U β r → r→∞ 0 weakly in L 2 (R n ) as well as in L 1 (R n ). Therefore, F (Φ 2 U β r )(ξ) → r→∞ 0 point-wise and in L 2 loc (R n ) ( in view of the bound |F (Φ 2 U β r )(ξ)| ≤ Φ 2 U β r 1 ≤ const ).
Taking into account that the function χ(ξ) = ψ(ξ) -ψ(π X (ξ)) is bounded and has a compact support, we conclude that

F (Φ 2 U β r )(ξ)χ(ξ) → r→∞ 0 in L 2 (R n ).
This implies that

lim r→∞ R n F (Φ 1 U α r )(ξ)F (Φ 2 U β r )(ξ)χ(ξ)dξ = 0. Therefore, lim r→∞ R n F (Φ 1 U α r )(ξ)F (Φ 2 U β r )(ξ) ψ(ξ)dξ = lim r→∞ R n F (Φ 1 U α r )(ξ)F (Φ 2 U β r )(ξ)ψ(π X (ξ))dξ = µ αβ , Φ 1 (x)Φ 2 (x)ψ(ξ) ,
as required.

Let the sequence U r = {U α r } N α=1 converges weakly as r → ∞ to the zero vector, let it be bounded in L p loc (Ω, R N ), p ≥ 2, and let µ = {µ αβ } N α,β=1 be an ultra-parabolic H-measure corresponding to this sequence. We define η = Trµ = N α=1 µ αα . As follows from Proposition 2, η is a locally finite non-negative measure on Ω × S X . We assume that this measure is extended on σ-algebra of η-measurable sets, and in particular that this measure is complete. We denote by γ the projection of η on Ω, that is,

γ(A) = η(A × S X ) if the set A × S X is η-measurable.
Obviously, γ is a complete locally finite measure on Ω, γ ≥ 0. Under the above assumptions the following statements hold.

Proposition 3. (i) As r → ∞ |U r | 2 = N α=1 |U α r (x)| 2 → γ weakly in M loc (Ω); if p > 2 then γ ∈ L p/2
loc (Ω) (here we identify γ and the corresponding density γ of γ with respect to the Lebesgue measure dx, so that γ = γ(x)dx), and |U r | 2 → γ(x) weakly in L Proof. By the Plancherel identity and relation [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] with ψ ≡ 1

Ω Φ 1 (x)Φ 2 (x)|U r | 2 dx = N α=1 R n F (Φ 1 U α r )(ξ)F (Φ 2 U α r )(ξ)dξ → r→∞ η(x, ξ), Φ 1 (x)Φ 2 (x) = γ, Φ 1 (x)Φ 2 (x) Since any function Φ(x) ∈ C 0 (Ω) can be represented in the form Φ(x) = Φ 1 (x)Φ 2 (x) ( for instance, one can take Φ 1 (x) = Φ(x), Φ 2 (x) being arbitrary function in C 0 (Ω) equal to 1 on supp Φ 1 (x) ), we conclude that |U r | 2 → γ as r → ∞ weakly in M loc (Ω). In the case p > 2 (here p/2 = ∞ if p = ∞) the sequence |U r | 2 is bounded in L p/2
loc (Ω), and we conclude that γ ∈ L p/2 loc (Ω). The first assertion is proved.

To prove (ii), remark firstly that µ αα ≤ η for all α = 1, . . . , N. Now, suppose that α, β ∈ {1, . . . , N}, α = β. By Proposition 2 for any compact set B ⊂ Ω×S X the matrix µ αα (B) µ αβ (B)

µ αβ (B) µ ββ (B)
is positive-definite; in particular,

|µ αβ (B)| ≤ µ αα (B)µ ββ (B) 1/2 ≤ η(B).
By regularity of measures µ αβ and η this estimate is satisfied for all Borel sets B. This easily implies the inequality Var µ αβ ≤ η. In particular, the measures µ αβ are absolutely continuous with respect to η, and by the Radon-Nykodim theorem µ αβ = h αβ (x, ξ)η, where the densities h αβ (x, ξ) are η-measurable and, as follows from the inequalities Var µ αβ ≤ η, |h αβ (x, ξ)| ≤ 1 η-a.e. on Ω × S X . We denote by H(x, ξ) the matrix with components h αβ (x, ξ). Recall that the H-measure µ is positive definite. This means that for all

ζ ∈ C N µζ • ζ = H(x, ξ)ζ • ζη ≥ 0. ( 18
) Hence H(x, ξ)ζ • ζ ≥ 0 for η-a.e. (x, ξ) ∈ Ω × S X . Choose a countable dense set E ⊂ C N . Since E is countable, then it follows from (18) that for a set (x, ξ) ∈ Ω × S X of full η-measure H(x, ξ)ζ • ζ ≥ 0 ∀ζ ∈ E,
and since E is dense we conclude that actually H(x, ξ)ζ • ζ ≥ 0 for all ζ ∈ C N . Thus, the matrix H(x, ξ) is Hermitian and positive definite for η-a.e. (x, ξ). After an appropriate correction on a set of null η-measure, we can assume that the above property is satisfied for all (x, ξ) ∈ Ω × S X , and also |h αβ (x, ξ)| ≤ 1 for all (x, ξ) ∈ Ω × S X , α, β = 1, . . . , N. The proof is complete.

Corollary 1. Suppose that the sequence U r = {U α r } N α=1 is bounded in L p loc (Ω, R N ), p > 2.
Let q = p/(p -2) (as usual we set q = 1 if p = ∞), and let L 2q 0 (Ω) be the space of functions in L 2q (Ω) having compact supports. Then relation [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] still holds for all functions Φ 1 (x), Φ 2 (x) ∈ L 2q 0 (Ω), ψ(ξ) ∈ C(S X ). Proof. Let K be a compact subset of Ω and Φ 1 (x), Φ(x) ∈ L 2q (K). The functions from L 2q (K) are supposed to be extended on Ω as zero functions outside of K. Using the Plancherel identity and the Hölder inequality (observe that

1 2q + 1 p = 1 2
), we get the following estimate

R n F (Φ 1 U α r )(ξ)F (Φ 2 U β r )(ξ)ψ(π X (ξ))dξ ≤ ψ ∞ Φ 1 U α r 2 Φ 2 U β r 2 ≤ (C K ) 2 ψ ∞ • Φ 1 2q Φ 2 2q , (19)
where

C K = sup r∈N U r L p (K)
. On the other hand, by Proposition 3

| µ αβ , Φ 1 (x)Φ 2 (x)ψ(ξ) | = | η, h αβ (x, ξ)Φ 1 (x)Φ 2 (x)ψ(ξ) | ≤ ψ ∞ Ω |Φ 1 (x)Φ 2 (x)|γ(x)dx ≤ ψ ∞ γ L p/2 (K) Φ 1 2q Φ 2 2q (20)
(in the last estimate we used again the Hölder inequality). Estimates (19), (20) show that both sides of relation ( 9) are continuous with respect to (Φ 1 , Φ 2 ) ∈ (L 2q (K)) 2 . Since (9) holds for Φ 1 , Φ 2 ∈ C 0 (K) and the space C 0 (K) is dense in L 2q (K), we claim that (9) holds for each Φ 1 (x), Φ 2 (x) ∈ L 2q (K). To conclude the proof, it only remains to notice that K is an arbitrary compact subset of Ω.

We will need in the sequel some results about Fourier multipliers in spaces

L d , d > 1. Recall that a function a(ξ) ∈ L ∞ (R n ) is a Fourier multiplier in L d if the pseudo-differential operator A with the symbol a(ξ), defined as F (Au)(ξ) = a(ξ)F (u)(ξ), u = u(x) ∈ L 2 (R n ) ∩ L d (R n ) can be extended as a bounded operator on L d (R n ), that is Au d ≤ C u d ∀u ∈ L 2 (R n ) ∩ L d (R n ), C = const.
We denote by M d the space of Fourier multipliers in L d . We also denote

Ṙn = (R \ {0}) n = { ξ = (ξ 1 , . . . , ξ n ) | n k=1 ξ k = 0 }.
The following statement readily follows from the Marcinkiewicz multiplier theorem (see [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF]Chapter 4]).

Theorem 2. Suppose that a(ξ) ∈ C n ( Ṙn ) is a function such that for some constant C

|ξ α D α a(ξ)| ≤ C ∀ξ ∈ Ṙn (21) for every multi-index α = (α 1 , . . . , α n ) such that |α| = α 1 + • • • + α n ≤ n. Then a(ξ) ∈ M d for all d > 1.
Here we use the standard notations

ξ α = n k=1 (ξ k ) α k , D α = n k=1 ∂ ∂ξ k α k .
Actually (see [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF]), it is sufficient to require that ( 21) is satisfied for multi-indexes α such that α k ∈ {0, 1}, k = 1, . . . , n.

We also need the following simple lemma (see [START_REF] Panov | Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property[END_REF]Lemma 8]).

Lemma 3. Let h(y, z) ∈ C n ((R ν × R n-ν ) \ {0}) be such that for some k ∈ N, γ ∈ R ∀t > 0 h(t k y, tz) = t γ h(y, z). ( 22
)
Then there exists a constant C > 0 such that for each multi-indexes α = (α 1 , . . . , α ν ), β = (β 1 , . . . , β n-ν ), |α| + |β| ≤ n and all y ∈ R ν , z ∈ R n-ν , y, z = 0

|D α y D β z h(y, z)| ≤ C(|y| 2 + |z| 2k ) γ 2k |y| -|α| |z| -|β| . Proof.
In view of ( 22), for all t > 0 we have

D α y D β z h(y, z) = t k|α|+|β|-γ (D α y D β z h)(t k y, tz).
Taking t = (|y| 2 + |z| 2k ) -1 2k in this relation, we find

D α y D β z h(y, z) = (|y| 2 + |z| 2k ) γ-k|α|-|β| 2k (D α y D β z h)(y ′ , z ′ ), (23)
where

y ′ = t k y, z ′ = tz, so that |y ′ | 2 + |z ′ | 2k = 1. Since the set of such (y ′ , z ′ ) is a compact subset of R n \ {0} the derivatives (D α y D β z h)(y ′ , z ′ ), |α| + |β| ≤ n,
are bounded on this set, and relation (23) implies that for some constant C > 0

|D α y D β z h(y, z)| ≤ C(|y| 2 + |z| 2k ) γ 2k (|y| 2 + |z| 2k ) -|α|/2 (|y| 2 + |z| 2k ) -|β|/(2k) ≤ C(|y| 2 + |z| 2k ) γ 2k |y| -|α| |z| -|β|
for all y, z = 0. The proof is complete.

Now we can prove that some useful for us functions are Fourier multipliers. Namely, assume that X is a linear subspace of R n , and let π X : R n → S X be the projection defined in Section 2.

Proposition 4 (cf. [5, Proposition 6]). The following functions are multipliers in spaces L d for all d > 1:

(i) a 1 (ξ) = ψ(π X (ξ)) where ψ ∈ C n (S X ); (ii) a 2 (ξ) = ρ(ξ)(1 + | ξ| 2 + | ξ| 4 ) 1/2 (| ξ| 2 + | ξ| 4 ) -1/2 , where ρ(ξ) ∈ C ∞ (R n ) is a function such that 0 ≤ ρ(ξ) ≤ 1, ρ(ξ) = 0 for | ξ| 2 + | ξ| 4 ≤ 1, ρ(ξ) = 1 for | ξ| 2 + | ξ| 4 ≥ 2; (iii) a 3 (ξ) = (1 + |ξ| 2 ) 1/2 (1 + | ξ| 2 + | ξ| 4 ) -1/2 ; (iv) a 4 (ξ) = (1 + | ξ| 2 + | ξ| 4 ) 1/2 (1 + |ξ| 2 ) -1 .
Proof. Since the space M d is invariant under non-degenerate linear transformations of the variables ξ ( see [START_REF] Bergh | Interpolation spaces. An introduction[END_REF]Chapter 6] ) then we can assume that To prove that a 2 (ξ) ∈ M d we introduce the function h 1 (s, y, z) = (s 2 + |y| 2 + |z| 4 ) 1/2 , s ∈ R. This function satisfies the assumptions of Lemma 3 with y replaced by (s, y) ∈ R ν+1 , and k = γ = 2. By this Lemma

X = R ν = {ξ ∈ R n | ξ = (y 1 , . . . , y ν , 0, . . . , 0) } while X ⊥ = {ξ ∈ R n | ξ = (0, . . . , 0, z 1 , . . . , z n-ν ) }. Since π X (t 2 y, tz) = π X (y, z) for t > 0, y ∈ X, z ∈ X ⊥ then h = a 1 (ξ) = ψ(π X (ξ))
|D α y D β z h 1 (s, y, z)| ≤ C(s 2 + |y| 2 + |z| 4 ) 1/2 |y| -|α| |z| -|β| , C = const.
Taking s = 1 in this relation, we arrive at the estimate

|D α y D β z h 1 (1, y, z)| ≤ C(1 + |y| 2 + |z| 4 ) 1/2 |y| -|α| |z| -|β|
, and by the Leibnitz formula we obtain that for each multi-indexes α, β such that

|α| + |β| ≤ n |D α y D β z ρ(y, z)h 1 (1, y, z)| ≤ C 1 (1 + |y| 2 + |z| 4 ) 1/2 |y| -|α| |z| -|β| , (24)
C 1 = const ( we use that ρ(y, z) = 1 for |y| 2 + |z| 4 ≥ 2 ). Let h 2 (y, z) = (|y| 2 + |z| 4 ) -1/2
. This function satisfies (22) with k = 2, γ = -2. By Lemma 3 for some constant C 2 and every multi-indexes α, β such that |α|

+ |β| ≤ n |D α y D β z h 2 (y, z)| ≤ C 2 (|y| 2 + |z| 4 ) -1/2 |y| -|α| |z| -|β| . ( 25
)
By the Leibnitz formula we derive from ( 24), (25) the estimates 22) where y is replaced by (s, y) ∈ R l+1 with the parameters k = γ = 1; k = 1, γ = -2; k = 2, γ = -2; k = γ = 2, respectively. By Lemma 3 we find that for each α, β, |α|

|D α y D β z ρ(y, z)h 1 (1, y, z)h 2 (y, z)| ≤ C 3 (1 + |y| 2 + |z| 4 ) 1/2 (|y| 2 + |z| 4 ) -1/2 |y| -|α| |z| -|β| ≤ 2C 3 |y| -|α| |z| -|β| (
(s, y, z) = (s 2 + |y| 2 + |z| 2 ) 1/2 , h 2 (s, y, z) = (s 2 +|y| 2 +|z| 2 ) -1 , h 3 (s, y, z) = (s 2 +|y| 2 +|z| 4 ) -1/2 , h 4 (s, y, z) = (s 2 +|y| 2 +|z| 4 ) 1/2 , s ∈ R, y ∈ X = R ν , z ∈ X ⊥ . These functions satisfy (
+ |β| ≤ n |y| |α| |z| |β| |D α y D β z h 1 (1, y, z)| ≤ C(1 + |y| 2 + |z| 2 ) 1/2 , |y| |α| |z| |β| |D α y D β z h 2 (1, y, z)| ≤ C(1 + |y| 2 + |z| 2 ) -1 , |y| |α| |z| |β| |D α y D β z h 3 (1, y, z)| ≤ C(1 + |y| 2 + |z| 4 ) -1/2 , |y| |α| |z| |β| |D α y D β z h 4 (1, y, z)| ≤ C(1 + |y| 2 + |z| 4 ) 1/2
, where C = const. Since a 3 (ξ) = h 1 (1, y, z)h 3 (1, y, z), a 4 (ξ) = h 2 (1, y, z)h 4 (1, y, z) where y = ξ, z = ξ then, using again the Leibnitz formula, we derive the estimates: for some constant C

|y| |α| |z| |β| |D α y D β z a 3 (y, z)| ≤ C(1 + |y| 2 + |z| 2 ) 1/2 (1 + |y| 2 + |z| 4 ) -1/2 ≤ 2C, |y| |α| |z| |β| |D α y D β z a 4 (y, z)| ≤ C(1 + |y| 2 + |z| 2 ) -1 (1 + |y| 2 + |z| 4 ) 1/2 ≤ 2C.
Here we take into account the following simple inequalities:

1 + |y| 2 + |z| 2 1 + |y| 2 + |z| 4 = 1 + |y| 2 1 + |y| 2 + |z| 4 + |z| 2 1 + |y| 2 + |z| 4 ≤ 1 + min(|z| 2 , |z| -2 ) ≤ 2, (1 + |y| 2 + |z| 4 ) 1/2 1 + |y| 2 + |z| 2 ≤ (1 + |y| 2 ) 1/2 1 + |y| 2 + |z| 2 + |z| 2 1 + |y| 2 + |z| 2 ≤ 2.
In view of Theorem 2, we conclude that a 3 (ξ), a 4 (ξ) ∈ M d for each d > 1. The proof is now complete.

We define the anisotropic Sobolev space W -1,-2

d consisting of distributions u(x) such that (1 + | ξ| 2 + | ξ| 4 ) -1/2 F (u)(ξ) = F (v)(ξ), v = v(x) ∈ L d (R n ). This
is a Banach space with the norm u = v d . The following proposition claims that this space lays between the spaces W -1

d and W -2 d . Proposition 5 (cf. [5, Proposition 7]). For each d > 1 W -1 d ⊂ W -1,-2 d ⊂ W -2 d
and the both embeddings are continuous.

Proof. Let u ∈ W -1 d . This means that (1 + |ξ| 2 ) -1/2 F (u)(ξ) = F (w)(ξ), w = w(x) ∈ L d (R n ). By Proposition 4(iii) a 3 (ξ) = (1+|ξ| 2 ) 1/2 (1+| ξ| 2 +| ξ| 4 ) -1/2 ∈ M d . Therefore, (1 + | ξ| 2 + | ξ| 4 ) -1/2 F (u)(ξ) = a 3 (ξ)F (w)(ξ) = F (v)(ξ), v(x) ∈ L d (R n ), that is, u ∈ W -1,-2 d . We deduce that W -1 d ⊂ W -1,-2 d . Since v d ≤ C w d , C = const this embedding is continuous. Now suppose that u ∈ W -1,-2 d . Then (1 + | ξ| 2 + | ξ| 4 ) -1/2 F (u)(ξ) = F (v)(ξ), v = v(x) ∈ L d (R n ). By Proposition 4(iv) a 4 (ξ) = (1 + | ξ| 2 + | ξ| 4 ) 1/2 (1 + |ξ| 2 ) -1 ∈ M d , and (1 + |ξ| 2 ) -1 F (u)(ξ) = a 4 (ξ)F (v)(ξ) = F (w)(ξ), w ∈ L d (R n ). This means that u ∈ W -2 d . We established that W -1,-2 d ⊂ W -2 d .
The continuity of this embedding follows from the estimate

w d ≤ C v d , C = const. The proof is complete.
We also introduce the local space W -1,-2 d,loc (Ω) consisting of distributions u(x) such that uf (x) belongs to W -1,-2 d for all f (x) ∈ C ∞ 0 (Ω). The space W -1,-2 d,loc (Ω) is a locally convex space with the topology generated by the family of semi-norms

u → uf W -1,-2 d , f (x) ∈ C ∞ 0 (Ω)
. Analogously, we define the spaces W -1 d,loc (Ω), W -2 d,loc (Ω). As it readily follows from Proposition 5,

W -1 d,loc ⊂ W -1,-2 d,loc
⊂ W -2 d,loc and these embeddings are continuous.

We will need also the following statement, which is rather well known (see, for example, [START_REF] Panov | Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property[END_REF]Lemma 6]).

Lemma 4. Let U r (x) be a sequence bounded in L 2 (R n ) ∩ L 1 (R n ) and weakly convergent to zero; let a(ξ) be a bounded function on R n such that a(ξ) → 0 as |ξ| → ∞. Then a(ξ)F (U r )(ξ) → r→∞ 0 in L 2 (R n ).
Proof. First, observe that by the assumption that a(ξ) → 0 at infinity, for any ε > 0 we can choose R > 0 such that |a(ξ

)| < ε for |ξ| > R. Then |ξ|>R |a(ξ)| 2 |F (U r )(ξ)| 2 dξ ≤ ε 2 F (U r ) 2 = ε 2 U r 2 ≤ Cε 2 , (27)
where C = sup r∈N U r 2 is a constant independent of r.

Further, by our assumption U r → 0 as r → ∞ weakly in L 1 . This implies that F (U r )(ξ) → 0 point-wise as r → ∞. Moreover, |F (U r )(ξ)| ≤ U r 1 ≤ const. Hence, using the Lebesgue dominated convergence theorem, we find that

|ξ|≤R |a(ξ)| 2 |F (U r )(ξ)| 2 dξ → 0 (28)
as r → ∞. It follows from ( 27), (28) that

lim r→∞ R n |a(ξ)| 2 |F (U r )(ξ)| 2 dξ ≤ Cε 2 .
Since ε > 0 is arbitrary, we conclude that

lim r→∞ R n |a(ξ)| 2 |F (U r )(ξ)| 2 dξ = 0, that is, a(ξ)F (U r )(ξ) → r→∞ 0 in L 2 (R n ).
The proof is complete.

Localization principle and proof of Theorem 1

Suppose that the sequence u r (x) converges weakly to u(x) in L p loc (Ω, R N ), and the sequences of distributions d,loc (Ω), where d > 1 is indicated in the Introduction. We will also assume that d ≤ 2. This assumption is not restrictive, because of the natural embeddings

W -1,-2 d,loc (Ω) ⊂ W -1,-2 d 1 ,loc (Ω) for each d 1 < d. Let U r = u r (x) -u(x) = (U 1 r , . . . , U N r ), U α r = u αr (x) -u α (x). Then U r → 0 as r → ∞ weakly in L 2 loc (Ω, R N ).
Therefore, after extraction of a subsequence (still denoted U r ), we can assume that the parabolic H-measure µ = {µ αβ } N α,β=1 corresponding to the subspace Proof.

X = R ν = { ξ = (ξ 1 , . . . , ξ ν , 0, . . . , 0) ∈ R n }
Since the coefficients a sαk (x), b sαkl (x) belong to L 2q loc (Ω), and 1 2q + 1 p = 1 2 , the sequences a sαk U αr , b sαkl U αr converge to zero as r → ∞ weakly in L 2 loc (Ω, R N ) and the sequences of distributions

L sr . = N α=1 n k=1 ∂ x k (a sαk U α r ) + N α=1 n k,l=ν+1 ∂ x k x l (b sαkl U α r ), r ∈ N, s = 1, . . . , m,
converge weakly to zero. Using the pre-compactness of these sequences in W -1,-2 d,loc (Ω), we find that L sr → 0 as r → ∞ in W -1,-2 d,loc (Ω). We choose Φ 1 (x) ∈ C ∞ 0 (Ω) and consider the distributions

l sr = ∂ x k (a sαk Φ 1 U α r -2b sαkl U α r ∂ x l Φ 1 ) + ∂ x k x l (b sαkl Φ 1 U α r ). ( 29
)
To simplify the notation, we use here and below the conventional rule of summation over repeated indexes, and suppose that the coefficients b sαkl are defined for all k, l = 1, . . . , n with b sαkl = 0 if min(k, l) ≤ ν. We can also assume that b sαkl = b sαlk for k, l = 1, . . . , n. Then, as it is easy to compute,

l sr = Φ 1 L sr + a sαk U α r ∂ x k Φ 1 -b sαkl U α r ∂ x k x l Φ 1 . (30)
Since the coefficients a sαk (x), b sαkl (x) belong to L 2q loc (Ω), and

1 2q + 1 p = 1 2 , the se- quences a sαk U α r ∂ x k Φ 1 , b sαkl U α r ∂ x k x l Φ 1 are bounded in L 2 (R n ).
Noticing that the function Φ 1 (x) has a compact support, we see that these sequences are bounded also in L d (R n ) for all s = 1, . . . , m, and they weakly converge to zero as r → ∞. Therefore, they converge to zero strongly in W -1 d (R n ) and, in view of Proposition 5, also in W -1,-2 d (R n ). By our assumptions, Φ 1 L sr → 0 as r → ∞ in W -1,-2 d (R n ). Hence, it follows from the above limit relations and (30) that l sr → 0 as r → ∞ in W -1,-2 d (R n ). Applying the Fourier transformation to this relation and then multiplying by (1 + | ξ| 2 + | ξ| 4 ) -1/2 , we arrive at

(1 + | ξ| 2 + | ξ| 4 ) -1/2 2πiξ k F (a sαk Φ 1 U α r )(ξ) - 4πiξ k F (b sαkl U α r ∂ x l Φ 1 )(ξ) -4π 2 ξk ξl F (b sαkl Φ 1 U α r )(ξ) = F (v sr )(ξ), (31)
where v sr → 0 as r → ∞ in L d (R n ). We take also into account that

ξ k ξ l F (b sαkl Φ 1 U α r )(ξ) = n k,l=ν+1 ξ k ξ l F (b sαkl Φ 1 U α r )(ξ) = ξk ξl F (b sαkl Φ 1 U α r )(ξ).
By Proposition 4(ii), we have

a 2 (ξ) = ρ(ξ)(1 + | ξ| 2 + | ξ| 4 ) 1/2 (| ξ| 2 + | ξ| 4 ) -1/2 ∈ M d .
Therefore, it follows from (31) that

ρ(ξ)(| ξ| 2 + | ξ| 4 ) -1/2 2πiξ k F (a sαk Φ 1 U α r )(ξ) -4πiξ k F (b sαkl U α r ∂ x l Φ 1 )(ξ) -4π 2 ξk ξl F (b sαkl Φ 1 U α r )(ξ) = a 2 (ξ)F (v sr )(ξ) = F (w sr )(ξ), ( 32
)
w sr → 0 as r → ∞ in L d (R n ) for all s = 1, . . . , m. Since ρ(ξ)| ξ| 2 (| ξ| 2 + | ξ| 4 ) 1/2 ≤ 1, ρ(ξ)|ξ| (| ξ| 2 + | ξ| 4 ) 1/2 ≤ ρ(ξ) | ξ| + | ξ| (| ξ| 2 + | ξ| 4 ) 1/2 ≤ 1 + min(| ξ|, | ξ| -1 ) ≤ 2 ( recall that 0 ≤ ρ(ξ) ≤ 1, and ρ(ξ) = 0 for | ξ| 2 + | ξ| 4 ≤ 1 ), and
F (a sαk Φ 1 U α r )(ξ), F (b sαkl Φ 1 U α r )(ξ), F (b sαkl U α r ∂ x l Φ 1 )(ξ) ∈ L 2 (R n ), we see that F (w sr )(ξ) ∈ L 2 (R n ), which implies that w sr ∈ L 2 (R n ) as well. Since b sαkl = 0 for k ≤ ν, ξk F (b sαkl U α r ∂ x l Φ 1 )(ξ) = ν k=1 ξ k F (b sαkl U α r ∂ x l Φ 1 )(ξ) = 0. (33)
Now, observe that for each k the function

a(ξ) = ρ(ξ) ξk (| ξ| 2 + | ξ| 4 ) 1/2
, satisfies the assumption of Lemma 4. Indeed, this follows from the estimate

|a(ξ)| ≤ ρ(ξ)(| ξ| 2 + | ξ| 4 ) -1/4 | ξ| (| ξ| 2 + | ξ| 4 ) 1/4 ≤ ρ(ξ)(| ξ| 2 + | ξ| 4 ) -1/4 . Since the sequences a sαk Φ 1 U α r , b sαkl U α r ∂ x l Φ 1 are bounded in L 2 (R n ) ∩ L 1 (R n
) and weakly converge to zero as r → ∞, then by Lemma 4

ρ(ξ) (| ξ| 2 + | ξ| 4 ) 1/2 ξk F (a sαk Φ 1 U α r )(ξ) → r→∞ 0 in L 2 (R n ), (34)
ρ(ξ) (| ξ| 2 + | ξ| 4 ) 1/2 ξk F (b sαkl U α r ∂ x l Φ 1 )(ξ) → r→∞ 0 in L 2 (R n ). (35)
Since H = R 2 then h αβ (x, ξ) = r αj r βj , where r ij = r ij (x, ξ), i, j = 1, . . . , N are components of matrix R. Therefore,

q αβ (x)h αβ = q αβ (x)r αj r βj = N j=1 Q(x)Re j • Re j , (40)
where {e j } N j=1 is the standard basis in C N . Since Re j ∈ Im R ⊂ Λ(x) then it follows from the assumption of Theorem 1 that Q(x)Re j • Re j ≥ 0 for η-a.e. (x, ξ) ∈ Ω × S X . In view of (40) we find that q αβ (x)h αβ (x, ξ) ≥ 0 for η-a.e. (x, ξ) ∈ Ω × S X . Now, it readily follows from (39) that lim r→∞ Ω (Φ(x)) 2 q(x, U r (x))dx ≥ 0 (41) for all real Φ(x) ∈ C 0 (Ω).

In view of the weak convergence u r → u, q(x, u r (x)) → v as r → ∞,

q(x, U r (x)) = q(x, u r (x)) + q(x, u(x)) -2 Re(Q(x)u r (x) • u(x)) → v -q(x, u(x))
weakly in M loc (Ω), and we derive from (41) that v -q(x, u(x))dx, (Φ(x)) 2 ≥ 0 Since (Φ(x)) 2 is an arbitrary nonnegative function in C 0 (Ω), this implies that q(x, u(x)) ≤ v. The proof is complete.

Corollary 3. Suppose that q(x, λ) = 0 for all λ ∈ Λ(x), x ∈ Ω. Then v = q(x, u(x)), that is, the functional u → q(x, u) is weakly continuous.

Proof. Applying Theorem 1 to the quadratic forms ±q(x, u), we obtain the inequalities ±v ≥ ±q(x, u(x)), which readily imply that v = q(x, u(x)).

Remark 2. In the particular case ν = n relations (2) are reduced to the requirement that the sequences of distributions

L sr = N α=1 n k=1 ∂ x k (a sαk (x)u αr ), s = 1, . . . , m are pre-compact in W -1 d,loc (
Ω). In applications to conservation laws, it usually happens that the sequences u αr are bounded in L ∞ loc (Ω) (so that p = ∞) while the sequences L sr are bounded in M loc (Ω). Since the space M loc (Ω) is compactly embedded in W -1 d,loc (Ω) for d < n/(n -1) then condition (2) is satisfied.

Then the sequence (u 1r , u 2r ) converges weakly to (u 1 , u 2 ) ∈ L p loc (Ω, R 2 ) with u 1 = u(t, x). Further, it satisfies the condition that the sequence of distributions

f r = ∂ t u 1r - n k,l=1 ∂ x k x l (a kl (t, x)u 2r ) is pre-compact in W -1,-2
d,loc (Ω). In accordance with (3), we define the set Λ = Λ(t, x):

Λ = {(λ 1 , λ 2 ) ∈ C 2 | ∃(ξ 0 , ξ) ∈ (R × R n ) \ {0} iξ 0 λ 1 + (A(t, x)ξ • ξ)λ 2 = 0 }.
Since (A(t, x)ξ • ξ) > 0 for ξ = 0 then Λ = { (λ 1 , λ 2 ) ∈ C 2 | Re λ 1 λ 2 = 0 }. Therefore, the quadratic functional q = q(u) = (u 1 u 2 + u 2 u 1 )/2 is zero for u = λ ∈ Λ. By Corollary 3 (observe that all the assumptions of this Corollary are satisfied) we claim that q(u 1r , u 2r ) = u 1r u 2r → r→∞ q(u 1 , u 2 ) = u 1 u 2 (42)

weakly in L 1 loc (Ω). Since the sequence u r is bounded in L p loc (Ω), p ≥ 2, then, extracting again a subsequence (still denoted by u r ), we may suppose that the Young measure ν t,x corresponding to this subsequence is well defined. Recall that a Young measure ν t,x on Ω is a weakly measurable map (t, x) → ν t,x of Ω into the space Prob(R) of probability measures on R. The weak measurability means that for each bounded continuous function p(λ) the function (t, x) → p(λ)dν t,x (λ) is Lebesgue measurable on Ω. It is known (see, for example, [START_REF] Pedregal | Parametrized measures and variational principles[END_REF]) that the Young measure corresponding to u r satisfies the property that whenever the sequence ψ(t, x, u r (t, x)) converges weakly in L 1 loc (Ω) for a Caratheodory function ψ(x, λ), its weak limit is the function ψ(t, x) = ψ(t, x, λ)dν t,x (λ). Moreover, ν t,x (λ) = δ(λ -u(t, x)), where δ(λ -u) is the Dirac mass at u, if and only if u r → u in L 1 loc (Ω). Since u r → u 1 = u(t, x), g(t, x, u r ) → u 2 (t, x), u r g(t, x, u r ) = u 1r u 2r → u 1 u 2 as r → ∞ weakly in L 1 loc (Ω) then these limit functions admit the representations: u 1 = λdν t,x (λ), u 2 = g(t, x, λ)dν t,x (λ), u 1 u 2 = λg(t, x, λ)dν t,x (λ).

It follows from these equalities that for a.e. (t, x) ∈ Ω u(t, x) g(t, x, λ)dν t,x (λ) = λg(t, x, λ)dν t,x (λ).

It is reduced to the equality (λ -u(t, x))g(t, x, λ)dν t,x (λ) = 0, and since (λ -u(t, x))ν t,x (λ) = 0, we arrive at the relation (λ -u(t, x))(g(t, x, λ) -g(t, x, u(t, x)))dν t,x (λ) = (λ -u(t, x))g(t, x, λ)dν t,x (λ)g(t, x, u(t, x)) (λ -u(t, x))ν t,x (λ) = 0 (43) for a.e. (t, x) ∈ Ω. Taking into account the fact that the function g(t, x, λ) is non-decreasing with respect to λ, we derive from (43) that for a.e. (t, x) ∈ Ω g(t, x, λ) = g(t, x, u(t, x)) on supp ν t,x . Therefore, u 2 = g(t, x, λ)dν t,x (λ) = g(t, x, u(t, x))

almost everywhere in Ω. Hence, in the limit as r → ∞

L(u r ) → L(u) = ∂ t u - n k,l=1
∂ x k x l (a kl (t, x)g(t, x, u)) in D ′ (Ω).

Since L(u r ) = f r → f as r → ∞ in D ′ (Ω), we conclude that L(u) = f . Besides, the image of ν t,x under the map u → g(t, x, u) coincides with the Dirac measure δ(λ -g(t, x, u(t, x))):

νt,x (λ) . = (g(t, x, •) * ν t,x )(λ) = δ(λ -g(t, x, u(t, x))).

It is easy to see that νt,x (λ) is the Young measure corresponding to the sequence g(t, x, u r (t, x)). Since this Young measure coincides with δ(λ -g(t, x, u(t, x))), we conclude that the sequence g(t, x, u r (t, x)) converges to g(t, x, u(t, x)) strongly in L p loc (Ω). Finally, observe that the limit function does not depend on the prescribed above choice of a subsequence. Therefore, g(t, x, u r (t, x)) also converges strongly to g(t, x, u(t, x)) for the original sequence u r . The proof is complete. Remark 3. In the case when the function g(t, x, u) is strictly monotone we deduce from Theorem 4 the strong pre-compactness property for weak solutions of the equation L(u) = f = f (t, x) ∈ W -1,-2 loc (Ω), which satisfy the equation in D ′ (Ω). Notice that for entropy solutions of this equation (with f = f (t, x, u) ∈ L 1 loc (Ω, C(R))) the strong pre-compactness property follows from general results of [START_REF] Panov | Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property[END_REF][START_REF] Panov | On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux[END_REF].

a

  sαk ∂ x k u αr , s = 1, . . . , m, r ∈ N,

  The H-measure µ is absolutely continuous with respect to η, more precisely, µ = H(x, ξ)η, where H(x, ξ) = {h αβ (x, ξ)} N α,β=1 is a bounded η-measurable function taking values in the cone of positive definite Hermitian N × N matrices, besides |h αβ (x, ξ)| ≤ 1.

  satisfies the assumptions of Lemma 3 with k = 2, γ = 0. By this Lemma for each multi-indexes α, β, |α| + |β| ≤ n |y| |α| |z| |β| |D α y D β z a 1 (y, z)| ≤ C = const. This, in particular, implies that assumption (21) of Theorem 2 is satisfied. By this Theorem we conclude that a 1 (ξ) ∈ M d for each d > 1.

∂∂

  x k (a sαk u αr ) + x k x l (b sαkl u αr ), r ∈ N, s = 1, . . . , m, are pre-compact in the anisotropic Sobolev space W -1,-2

is well defined. Theorem 3 (

 3 localization principle). For each s = 1, . . . , m; β = 1, . . . , N N α=1 P sα (x, ξ)µ αβ = 0, where P sα (x, ξ) = 2πi ν k=1 a sαk (x)ξ k -4π 2 n k,l=ν+1 b sαkl (x)ξ k ξ l .

  26) in the domain |y| 2 + |z| 4 ≥ 1, here |α| + |β| ≤ n, C 3 = const. In view of (26) we conclude that in this domain for each α, β, |α| + |β| ≤ n |y| |α| |z| |β| |D α y D β z a 2 (y, z)| ≤ const. Since a 2 (y, z) = 0 for |y| 2 + |z| 4 < 1 we see that the requirements of Theorem 2 are satisfied. Therefore, a 2 (ξ) ∈ M d for all d > 1. Now we introduce the functions h 1

Acknowledgements. This work was carried out during the author stay at the University of Franche-Comté (Besançon, France). The authors thanks colleagues from the Laboratory of Mathematics and especially Boris Andreianov for hospitality and fruitful discussions on the subject of this paper.

The author also acknowledges the support of the Russian Foundation for Basic Research (grant No. 09-01-00490-a) and and Deutsche Forschungsgemeinschaft (DFG project No. 436 RUS 113/895/0-1).

It follows from (33), (35) that

Let Φ 2 (x) ∈ C 0 (R n ), ψ(ξ) ∈ C n (S X ). Since the sequence Φ 2 U β r is bounded in L p (Ω) and supported in the compact supp Φ 2 , and d ′ = d/(d -1) ≤ p, this sequence is also bounded in L 2 (R n ) ∩ L d ′ (R n ), By Proposition 4(i) for a fixed β = 1, . . . , N ψ(π X (ξ))F (Φ 2 U β r )(ξ) = F (g r )(ξ), where the sequence g r is bounded in

and integrate the result over ξ ∈ R n . Passing then to the limit as r → ∞ and taking into account relations (34), (36), we arrive at

On the other hand, by relation ( 9), Remark 1 and Corollary 1 (in the case p > 2), we see that

Then it follows from (37) that

where

We underline that the functions P sα (x, ξ)Φ 1 (x)Φ 2 (x)ψ(ξ) are measurable and locally integrable with respect to the measure η. This is evident in the case p = 2 (then a sαk , b sαkl ∈ C(Ω)) while in the case p > 2 this readily follows from Proposition 3, from the assumptions a sαk , b sαkl ∈ L 2q loc (Ω), and from the inequality

) are arbitrary, we derive from (38) that P sα (x, ξ)µ αβ = 0 for each s = 1, . . . , m, β = 1, . . . , N. The proof is complete.

By Proposition 3 the H-measure µ admits the representation µ = H(x, ξ)η, where

Proof. By Theorem 3 P sα (x, ξ)h αβ (x, ξ)η = 0. This can be written as P (x, ξ)H(x, ξ) = 0, where P (x, ξ) is a m × N matrix with components P sα . Therefore, for η-a.e. (x, ξ) ∈ Ω × S X Im H(x, ξ) ⊂ ker P (x, ξ). Now notice that if λ = (λ 1 , . . . , λ N ) ∈ C N belongs to ker P (x, ξ) then

for all s = 1, . . . , m. Remark that 2πξ = 0 because of the inclusion ξ ∈ S X . Hence, λ ∈ Λ(x). We conclude that ker P (x, ξ) ⊂ Λ(x), and Im H(x, ξ) ⊂ ker P (x, ξ) ⊂ Λ(x), as was to be proved. Now we are ready to prove our main Theorem 1. Proof of Theorem 1.

Since H = H(x, ξ) ≥ 0 there exists a unique Hermitian matrix R = R(x, ξ) = H 1/2 such that R ≥ 0 and H = R 2 . By the known properties of Hermitian matrices ker R = ker H, which readily implies that Im R = Im H. By Corollary 2 we claim that Im R(x, ξ) ⊂ Λ(x) for η-a.e. (x, ξ) ∈ Ω×S X . Now we represent the coefficients q αβ (x) of quadratic form q(x, u) in the form q αβ (x) = q (1)

αβ (x), where for j = 1, 2 q (j)

For instance, we can set q (1)

Taking into account Corollary 1, we find that for real Φ(x) ∈ C 0 (Ω)

In the case ν = 0 the statement of Theorem 1 is a compensated compactness result under the second order constraints

which are required to be pre-compact in W -2 d,loc (Ω). Observe also that in each of the cases ν = n, 0 the set Λ(x) may be defined as a subset of real space R N .

Some applications

We consider the parabolic operator

, where q = p/(p -2), p > 2, and a kl ∈ C(Ω) if p = 2.

The matrix A(t, x) = {a kl (t, x)} n k,l=1 is supposed to be symmetric and strictly positive: A(t, x)ξ •ξ > 0 ∀ξ ∈ R n , ξ = 0. The function g(t, x, u) is a Caratheodory function on Ω × R, non-strictly increasing with respect to the variable u.

Assume that the sequences u r (t, x), g(t, x, u r (t, x)), r ∈ N are bounded in L p loc (Ω), moreover, if p = 2 assume that the sequence ρ(u r (t, x)g(t, x, u r (t, x))) is bounded in L 1 loc (Ω) for some positive super-linear function ρ

Ω), where the latter space correspond to the subspace X = { (ξ 0 , 0, . . . , 0) } ⊂ R n+1 , here (ξ 0 , ξ 1 , . . . , ξ n ) are the dual variables ( ξ 0 correspond to the time variable t ), and d = p/(p -1) (d > 1 in the case p = ∞). Theorem 4. Under the above assumptions, L(u) = f in D ′ (Ω). In addition, the sequence g(t, x, u r (t, x)) converges to g(t, x, u(t, x)) as r → ∞ strongly in L p loc (Ω).

Proof. Let u 1r = u r (t, x), u 2r = g(t, x, u r (t, x)). Passing to a subsequence if necessary, we can assume that u 2r (t, x) → u 2 = u 2 (t, x) weakly as r → ∞.