Ultra-parabolic equations with rough coefficients. Entropy solutions and strong pre-compactness property E.Yu. Panov 1

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1 Ultra-parabolic equations with rough coefficients. Entropy solutions and strong pre-compactness property E.Yu. Panov 1 Abstract Under some non-degeneracy condition we show that sequences of entropy solutions of a semi-linear ultra-parabolic equation are strongly pre-compact in the general case of a Caratheodory flux vector and a diffusion matrix. The proofs are based on localization principles for the parabolic H-measures corresponding to sequences of measure-valued functions. 1 Introduction Let be an open subset of R n. In the domain we consider the semi-linear ultraparabolic equation divϕ(x, u) D 2 B(x, u) + ψ(x, u) = 0, (1) where D 2 B(x, u) = x 2 i x j b ij (x, u), u = u(x) (we use the conventional rule of summation over repeated indexes), B(x, u) = {b ij (x, u)} n i,j=1 is a symmetric matrix. We shall assume that the components of this matrix are Caratheodory functions: b ij (x, u) L 2 loc (, C(R)), i, j = 1,..., n. This means that b ij(x, u) are measurable with respect to x, continuous with respect to u, and max b ij(x, u) L 2 loc () M > 0. In this case u M the parabolicity of (1) is understood in the following sense x, u 1, u 2 R, u 1 > u 2 B(x, u 1 ) B(x, u 2 ) 0, (2) that is, ξ R n (B(x, u 1 ) B(x, u 2 ))ξ ξ 0 ( here u v denotes the scalar product of vectors u, v R n ). We shall also assume that the matrix B(x, u) is degenerated on a linear subspace X R n, that is, for all ξ X the function B(x, u)ξ ξ does not depend on u: B(x, u)ξ ξ = C(x). Hence, (1) is a semi-linear ultra-parabolic equation. Concerning the convective terms, we suppose that ϕ(x, u) = (ϕ 1 (x, u),..., ϕ n (x, u)) L 2 loc (, C(R, Rn )) is a Caratheodory vector. We also assume that for any p R the distribution div x ϕ(x, p) D 2 x B(x, p) = γ p M loc (), (3) where M loc () is the space of locally finite Borel measures on with the standard locally convex topology generated by semi-norms p Φ (µ) = Var (Φµ), Φ = Φ(x) 1 The work was supported by the Russian Foundation for Basic Research (grant No a) and DFG project No. 436 RUS 113/895/0-1. This article was written as part of the the international research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year

2 C 0 (). The function ψ(x, u) is assumed to be a Caratheodory function on R: ψ(x, u) L 1 loc (, C(R)). Let γ p = γp r +γp s be the decomposition of the measure γ p into the sum of regular and singular measures, so that γp r = ω p (x)dx, ω p (x) L 1 loc (), and γs p is a singular measure (supported on a set of zero Lebesgue measure). We denote by γp s the variation of the measure γp, s which is a non-negative locally finite Borel measure on. Denote, as 1, u > 0, usual, sign u = 1, u < 0, 0, u = 0. Now, we introduce a notion of entropy solution of (1). Definition 1. A measurable function u(x) on is called an entropy solution of equation (1) if ϕ i (x, u(x)), b ij (x, u(x)), ψ(x, u(x)) L 1 loc (), i, j = 1,..., n, and for all p R the Kruzhkov-type entropy inequality (see [10]) holds div [sign(u(x) p)(ϕ(x, u(x)) ϕ(x, p))] D 2 (sign(u(x) p)(b(x, u(x)) B(x, p))) + sign(u(x) p)[ω p (x) + ψ(x, u(x))] γ s p 0 (4) in the sense of distributions on (in the space D ()); that is, for all non-negative functions f(x) C0 () sign(u(x) p)[(ϕ(x, u(x)) ϕ(x, p)) f(x) + (B(x, u(x)) B(x, p)) D 2 f (ω p (x) + ψ(x, u(x)))f(x)]dx + f(x)d γp (x) s 0. We use the notation D 2 f for the matrix { 2 x i x j f} n i,j=1 and P Q = TrP Q = n p ij q ij denotes scalar product of symmetric matrices P = {p ij } n i,j=1, Q = {q ij } n i,j=1. particular, (B(x, u(x)) B(x, p)) D 2 f = (b ij (x, u) b ij (x, p)) 2 x i x j f. In the case when the second-order term is absent ( B(x, u) 0 ) our definition extends the notion of the entropy solution for first-order balance laws introduced for the case of one space variable in [7, 8]. If ϕ(x, u), B(x, u) are smooth, and the strong ellipticity condition A(x, u) = B u(x, u) εe, ε > 0 is satisfied then weak (variational) solutions of (1) are entropy solutions as well. This fact will be demonstrated in last Section 5.2 ( as a part of the proof of Theorem 2 ). We also notice that we do not require u(x) to be a distributional solution of (1). If u(x) L () and γ s p = 0 for all p R then any entropy solution u(x) satisfies (1) in D (), i.e. u(x) is a distributional solution of (1). Indeed, this follows from (4) with p = ± u. But, generally, entropy solutions are not distributional ones, even in the case when the singular measures γ s p are absent. For instance, as is easily verified, 2 i,j=1 In

3 u(x) = sign x x 1/2 is an entropy solution of the first-order equation (xu 2 ) x = 0 on the line = R, but it does not satisfy this equation in D (R). We assume that equation (1) is non-degenerate in the sense of the following definition. Definition 2. Equation (1) is said to be non-degenerate if for almost all x for all ξ X, ξ X such that ξ 0, ξ 0 the function λ ξ ϕ(x, λ), λ B(x, λ) ξ ξ are not constant on non-degenerate intervals. In this paper, we shall establish the strong pre-compactness property for sequences of entropy solutions. This result generalizes the previous results of [12, 13, 14, 15, 16] to the case of ultra-parabolic equations. Theorem 1. Suppose that u k, k N is a sequence of entropy solutions of non-degenerate equation (1) such that ϕ(x, u k (x)) + ψ(x, u k (x)) + B(x, u k (x)) + m(u k (x)) is bounded in L 1 loc (), where m(u) is a nonnegative super-linear function (i.e. m(u)/u as u ). Then there exists a subsequence of u k, which converges in L 1 loc () to some entropy solution u(x). We use here and everywhere below the notation B for the Euclidean norm of a symmetric matrix B, that is B 2 = B B = TrB 2. More generally, we establish the strong pre-compactness of approximate sequences u k (x) for non-degenerate equation (1). The only assumption we need is that the sequences of distributions divϕ(x, s a,b (u k (x))) D 2 B(x, s a,b (u k (x))) are pre-compact in the anisotropic Sobolev space W 1, 2 d,loc () for some d > 1 and each a, b R, a < b where s a,b (u) = max(a, min(u, b)) are cut-off functions, and the space () will be specified below, in Section 4. We do not require here that condition (3) is satisfied. Remark that the non-degeneracy condition is essential for the statement of Theorem 1. For example, assume that (1) has the form divϕ(u) D 2 B(u) = 0 and ξ ϕ(u) = const on the segment [a, b] with ξ X, ξ 0 then the sequence u k (x) = [a + b + (b a) sin(kξ x)]/2 of entropy solutions does not contain strongly convergent subsequences. We also stress that for sequences of distributional solutions ( without additional entropy constraints ) the statement of Theorem 1 does not hold. For example, the sequence u k = sign sin kx consists of distributional solutions for the Burgers equation u t + (u 2 ) x = 0 ( as well as for the corresponding stationary equation (u 2 ) x = 0 ) and converges only weakly, while the non-degeneracy condition is evidently satisfied. W 1, 2 d,loc Theorem 1 will be proved in the last section. The proof is based on general localization properties for parabolic H-measures corresponding to bounded sequences of measure-valued functions. It also follows from these properties the strong convergence of various approximate solutions for equation (1). We describe below one useful approximation procedure. For simplicity we assume that ψ(x, u) 0. Let ζ(s) C0 (R) be a non-negative function with support in 3

4 the segment [ 1, 1] such that ζ(s)ds = 1. We set ζ m (s) = mζ(ms) for m N, n α m (y) = ζ m (y i ), y R n, so that the sequence α m is an approximate unity on R n. i=1 We introduce the averaged functions ϕ m (x, u) = (ϕ(, u) α m )(x) = ϕ(x y, u)α m (y)dy, R n B m (x, u) = (B(, u) α m )(x) = B(x y, u)α m (y)dy. R n Then, by known properties of averaged functions, ϕ m (x, u) C (, C(R, R n )), B m (x, u) C (, C(R, Sym n )), where Sym n denotes the space of symmetric matrices of order n, γ m (x, p). = div x ϕ m (x, p) D 2 x B m (x, p) C (, C(R)) for all p R, and ϕ m (x, ) ϕ(x, ) in L 2 loc(, C(R, R n )), (5) B m (x, ) B(x, ) in L 2 loc(, C(R, Sym n )), (6) γ m (x, p) γ p weakly in M loc (). (7) Then, recall that γ p = γ r p + γ s p, where γ r p = ω p (x)dx and therefore γ m (x, p) = (γ p α m )(x) = γ r mp + γ s mp, where γ r mp = ω p α m, γ s mp = γ s p α m C (), and γ mp r ω p in L 1 loc(), (8) γ mp s γp s α m γ s p weakly in M loc (). (9) Now we average the vector ϕ m and the matrix B m with respect to the variable u, introducing for l N the functions ϕ m,l (x, u) = ( ϕ m (x, ) ζ l )(u) = ϕ m (x, u v)ζ l (v)dv, B m,l (x, u) = ( B m (x, ) ζ l )(x) = B m (x, u v)ζ l (v)dv. Clearly, ϕ m,l (x, u) C ( R, R n ), B m,l (x, u) C ( R, Sym n ) and for each fixed m N ϕ m,l (x, u) ϕ m (x, u), Bm,l (x, u) Bm (x, u), l l div x ϕ m,l (x, u) Dx 2 B m,l (x, u) = γ m (x, ) ζ l (u) γ m (x, u) l 4

5 uniformly on compact subset of R. These relations allow to choose an increasing sequence l = l m in such a way that for ϕ m (x, u) =. ϕ m,lm (x, u), B m (x, u) =. B m,lm (x, u)+ ε m ue, where a sequence ε m > 0, ε m 0, and E is the unit matrix, we have ϕ m (x, u) ϕ m (x, u) 0, B m (x, u) B m (x, u) 0, (10) (div x ϕ m (x, u) Dx 2 B m (x, u)) (div x ϕ m (x, u) Dx 2 B m (x, u)) 0 (11) uniformly on compact subset of R. It follows from relations (10), (5), (6) that ϕ m (x, ) ϕ(x, ) in L 2 loc(, C(R, R n )), (12) B m (x, ) B(x, ) in L 2 loc(, C(R, Sym n )). (13) Now, observe that γ m (x, p). = div x ϕ m (x, p) D 2 x B m (x, p) = γ r mp + γ s mp, where γ r mp. = (div x ϕ m (x, p) D 2 x B m (x, p)) (div x ϕ m (x, p) Dx 2 B m (x, p)) + γ mp r ω p (x) in L 1 loc() (14) in accordance with (11), (8). Further, from relation (9) it follows that for each f(x) C 0 (), f(x) 0 f(x) γ mp(x) dx s f(x)d γp (x). s (15) lim Remark that, as follows from the assumption (2) and the choice of our approximations, A m (x, u) = (B m ) u(x, u) ε m E, and (A m (x, u) ε m E)ξ = 0 ξ X. Let K be a compact subset of, M > 0. We introduce the sequence M I m (K, M) = 1 + div x ϕ m (x, p) Dx 2 B m (x, p) dpdx. K M Generally, the sequence I m (K, M) may tend to infinity as m. Obviously, this sequence does not depend on ε m, which allows to choose the sequence ε m > 0 in such a way that ε m I m (K, M) 0 (16) for each M > 0 and each compact K. Now, we consider the approximate equation divϕ m (x, u) D 2 B m (x, u) = div[ ϕ m (x, u) A m (x, u) u] = 0, (17) where ϕ m (x, u) is a vector with coordinates ϕ mi (x, u) = ϕ mi (x, u) xj (B m ) ij (x, u), i = 1,..., n, 5

6 where ϕ mi (x, u), (B m ) ij (x, u), i, j = 1,..., n, are components of the vectors ϕ m (x, u) and the matrix B m (x, u), respectively. We suppose that u = u m (x) is a bounded weak solution of elliptic equation(17) ( for instance, we can take u = u m (x) being a weak solution to the Dirichlet problem with a bounded data at ). This means ( see [11][Chapter 4] ) that u L () W2,loc 1 (), where W 2,loc 1 () is the Sobolev space consisting of functions whose generalized derivatives lay in L 2 loc (), and the following standard integral identity is satisfied: f = f(x) C0(). 1 [ ϕ m (x, u(x)) A m (x, u(x)) u(x)] f(x)dx = 0. (18) We also assume that the sequence u m is bounded in L (). Under the above assumptions we establish the strong convergence of the approximations. Theorem 2. Suppose that equation (1) is non-degenerate. Then the sequence u m (x) u(x) in L 1 loc (), where u = u(x) is an entropy and a distributional solution of (1). Remark that Theorem 2 allows to establish the existence of entropy solutions of boundary value problems for equation (1) ( as well as initial or initial boundary value problems for evolutionary equations of the kind (1) ). For example, in [16] we use approximations and the strong pre-compactness property in order to prove the existence of entropy solutions to the Cauchy problem for an evolutionary hyperbolic equation with discontinuous multidimensional flux. This extends results of [9], where the two-dimensional case is treated by the compensated compactness method. In the next section 2 we describe the main concepts, in particular the concept of measure-valued functions. In sections 3,4 we introduce a notion of H-measure and prove the localization property. Finally, in the last section 5, these results are applied to prove our main Theorems 1,2. 2 Main concepts Recall ( see [3, 4, 20] ) that a measure-valued function on is a weakly measurable map x ν x of the set into the space of probability Borel measures with compact support in R. The weak measurability of ν x means that for each continuous function f(λ) the function x f(λ)dν x (λ) is Lebesgue-measurable on. Remark 1. If ν x is a measure-valued function then, as was shown in [13], the functions g(λ)dν x (λ) are measurable in for all bounded Borel functions g(λ). More generally, if f(x, λ) is a Caratheodory function and g(λ) is a bounded Borel function then the function f(x, λ)g(λ)dν x (λ) is measurable. This follows from the fact that any Caratheodory function is strongly measurable as a map x f(x, ) C(R) (see [6], Chapter 2) and, therefore, is a pointwise limit of step functions f m (x, λ) = 6

7 g mi (x)h mi (λ) with measurable functions g mi (x) and continuous h mi (λ) so that for i x f m (x, ) f(x, ) in C(R). A measure-valued function ν x is said to be bounded if there exists M > 0 such that supp ν x [ M, M] for almost all x. We denote by ν x the smallest value of M with this property. Finally, measure-valued functions of the form ν x (λ) = δ(λ u(x)), where δ(λ u) is the Dirac measure concentrated at u are said to be regular; we identify them with the corresponding functions u(x). Thus, the set M V () of bounded measure-valued functions on contains the space L (). Note that for a regular measure-valued function ν x (λ) = δ(λ u(x)) the value ν x = u. Extending the concept of boundedness in L () to measure-valued functions, we shall say that a subset A of MV () is bounded if sup νx A ν x <. Below we define the weak and the strong convergence of sequences of measurevalued functions. Definition 3. Let νx k MV (), k N, and let ν x MV (). Then 1) the sequence νx k converges weakly to ν x if for each f(λ) C(R), f(λ)dνx(λ) k f(λ)dν x (λ) weakly- in L (); k 2) the sequence νx k converges to ν x strongly if for each f(λ) C(R), f(λ)dνx(λ) k f(λ)dν x (λ) in L 1 k loc(). The next result was proved in [20] for regular functions νx. k The proof can be easily extended to the general case, as was done in [13]. Theorem T. Let νx, k k N be a bounded sequence of measure-valued functions. Then there exist a subsequence νx r = νx, k k = k r, and a measure-valued function ν x MV () such that νx r ν x weakly as r. Theorem T shows that bounded sets of measure-valued functions are weak1y precompact. If u k (x) L () is a bounded sequence, treated as a sequence of regular measure valued functions, and u k (x) weakly converges to a measure valued function ν x then ν x is regular, ν x (λ) = δ(λ u(x)), if and only if u k (x) u(x) in L 1 loc () ( see [20] ). Obviously, if u k (x) converges to ν x strongly then u k (x) u(x) = λdν x (λ) in L 1 loc () and then ν x(λ) = δ(λ u(x)). We shall study the strong pre-compactness property using Tartar s techniques of H-measures. Let F (u)(ξ) = e 2πiξ x u(x)dx, ξ R n, be the Fourier transform extended as unitary operator on the space u(x) L 2 (R n ), and let S = S n 1 = { ξ R n ξ = 1 } be the unit sphere in R n. Denote by u u, u C, the complex conjugation. The concept of H-measure corresponding to some sequence of vector-valued functions bounded in L 2 () was introduced by L. Tartar [21] and P. Gerárd [5] on the 7

8 basis of the following result. For l N let U k (x) = ( U 1 k (x),..., U l k (x)) L 2 (, R l ) be a sequence weakly convergent to the zero vector. Proposition 1. [see [21], Theorem 1.1] There exists a family of complex Borel measures µ = {µ ij } l i,j=1 in S and a subsequence U r(x) = U k (x), k = k r, such that ( ) ξ µ ij, Φ 1 (x)φ 2 (x)ψ(ξ) = lim F (U i rφ 1 )(ξ)f (Ur j Φ 2 )(ξ)ψ dξ (19) R ξ n for all Φ 1 (x), Φ 2 (x) C 0 () and ψ(ξ) C(S). The family µ = {µ ij } l i,j=1 is called an H-measure corresponding to U r(x). Recently in [1] the new concept of parabolic H-measures was introduced. Here we present the more general variant of this concept. 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 = P1 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 the projection π X : R n \ {0} S X : π X (ξ) = ξ ( ξ 2 + ξ 4 ) + ξ 1/2 ( ξ 2 + ξ 4 ). 1/4 Remark that in the case when X = {0} or X = R n π X (ξ) = ξ/ ξ. We denote p(ξ) = ( ξ 2 + ξ 4 ) 1/4. The following useful property of the projection holds. 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. We define for ξ R n, α > 0 ξ α = α 2 ξ + α ξ. Observe that for all α > 0 π X (ξ α ) = π X (ξ). Without lose 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 ( ) 1/2 ( α 4 ξ η 2 + α 2 ξ η 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. (20) Here we take into account that α 1 and therefore α 4 α 2. Since we have the estimate (β + α) 2 4β 2 = 4( η 2 + η 4 ) 1/2 4/ η, (β + α) 2 η 2 + η 2 4( η + η 2 ) 4 ( 2( η 2 + η 4 ) ) 1/2 6(p(η)) 2. (21) 8

9 Concerning the term β α, we estimate it as follows β α = p(ξ) p(η) p(ξ)p(η) = 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 ) ξ η p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) (p(ξ)) 2 + (p(η)) 2 + (p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) ξ η p(ξ)p(η)(p(ξ) + p(η))((p(ξ)) 2 + (p(η)) 2 ) 1 + p(ξ) + p(η) ξ η 2 ξ η p(ξ) + p(η) p(ξ)p(η) p(ξ)p(η). (22) 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 (20), (21), (22) that π X (ξ) π X (η) ξ η p(ξ) 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 ). Then we can define 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 [21], in the case when S X = S, π X (ξ) = ξ/ ξ the commutator [A, B] = AB BA is a compact operator. Using the assertion of Lemma 1 one can easily extend this result for the general case ( in the case dim X = 1 this was done in [1] ). For completeness we give below the details for the general setting. 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) C0 (R n ), and as k a k (z) a(z), b k (x) b(x) 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 ] [A, B] and it is sufficient to prove that the k 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. 9

10 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 ξ η a k (π X (ξ)) a k (π X (η)) C 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 ξ η. (23) m 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(ξ, η)(a k (π X (ξ)) a k (π X (η)))f (b k )(ξ η), r m (ξ, η) = (1 χ(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 (23) R m v(ξ) C (ξ η)f (b k )(ξ η) v(η) dη = [ ξf (b k ) v ](ξ) m R n 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 complete the proof. The parabolic H-measure µ ij, i, j = 1,..., l corresponding to a subspace X R n and a sequence U r (x) L 2 (, R l ) is defined on S X by the relation similar to (19): Φ 1 (x), Φ 2 (x) C 0 (), ψ(ξ) C(S X ) µ ij, Φ 1 (x)φ 2 (x)ψ(ξ) = lim F (Φ 1 U r)(ξ)f i (Φ 2 Ur j )(ξ)ψ(π X (ξ))dξ. (24) R n The existence of the H-measure µ ij is proved exactly in the same way as in [21], with using the statement of Lemma 2. This H-measure satisfies the same properties as the usual H-measure µ pq (corresponding to the case X = {0} or X = R n ). The concept of H-measure was extended in [13] ( see also [14, 15] ) to the case of continuous indexes i, j. The similar extension can be also established for parabolic H-measures. We study the properties of such H-measures in the next section. 10

11 3 H-measures corresponding to bounded sequences of measure-valued functions Let ν k x MV () be a bounded sequence of measure-valued functions weakly convergent to a measure-valued function ν 0 x MV (). For x and p R we introduce the distribution functions u k (x, p) = ν k x((p, + )), u 0 (x, p) = ν 0 x((p, + )). Then, as mentioned in Remark 1, for k N {0} and p R the functions u k (x, p) are measurable in x ; thus, u k (x, p) L () and 0 u k (x, p) 1. Let { } E = E(νx) 0 = p 0 R u 0 (x, p) u 0 (x, p 0 ) in L 1 p p0 loc(). We have the following result, whose proof can be found in [13]. Lemma 3. The complement Ē = R \ E is at most countable and if p E then u k (x, p) u 0 (x, p) weakly- in L (). k Let U p k (x) = u k(x, p) u 0 (x, p). Then, by Lemma 3, U p k (x) 0 as k weakly- in L () for p E. Let X be a linear subspace of R n. The next result, similar to Proposition 1, was also established in [13] in the case X = R n. The general case of arbitrary X is proved exactly in the same way. Proposition 2. 1) There exists a family of locally finite complex Borel measures {µ pq } p,q E in S X and a subsequence U r (x) = {Ur p (x)} p E, Ur p (x) = U p k (x), k = k r such that for all Φ 1 (x), Φ 2 (x) C 0 () and ψ(ξ) C(S X ) µ pq, Φ 1 (x)φ 2 (x)ψ(ξ) = lim F (Φ 1 Ur p )(ξ)f (Φ 2 Ur q )(ξ)ψ(π X (ξ))dξ. (25) R n 2) The correspondence (p, q) µ pq is a continuous map from E E into the space M loc ( S). We call the family of measures {µ pq } p,q E the H-measure corresponding to the subsequence νx r = νx, k k = k r. Remark 2. We can replace the function ψ(π X (ξ)) in relation (25) ( and in (24) ) to a function ψ(ξ) C(R n ), which equals ψ(π X (ξ)) for large ξ. Indeed, since Ur q 0 weakly- in L () we have F (Φ 2 Ur q )(ξ) 0 point-wise and in L 2 loc (Rn ) ( in view of the bound F (Φ 2 Ur q )(ξ) Φ 2 Ur q 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 This implies that F (Φ 2 U q r )(ξ)χ(ξ) 0 in L 2 (R n ). lim F (Φ 1 Ur p )(ξ)f (Φ 2 Ur q )(ξ)χ(ξ)dξ = 0. R n 11

12 Therefore lim F (Φ 1 Ur p )(ξ)f (Φ 2 Ur q )(ξ) ψ(ξ)dξ = R n lim F (Φ 1 Ur p )(ξ)f (Φ 2 Ur q )(ξ)ψ(π X (ξ))dξ = µ pq, Φ 1 (x)φ 2 (x)ψ(ξ), R n as required. We point out the following important properties of an H-measure. Lemma 4. (i) µ pp 0 for each p E; (ii) µ pq = µ qp for all p, q E; (iii) for p 1,..., p l E and g 1,..., g l C 0 ( S X ) the matrix A = a ij = µ p ip j, g i g j, i, j = 1,..., l is Hermitian and positive-definite. Proof. We prove (iii). First let the functions g i = g i (x, ξ) be finite sums of functions of the form Φ(x)ψ(ξ), where Φ(x) C 0 () and ψ(ξ) C(S X ). Then it follows from (25) that a ij = lim H r(ξ)h i r(ξ)dξ, j (26) R n where Hr(ξ) i = F (g i (, π X (ξ))u p i r )(ξ). Hence, setting g i (x, ξ) = g(x, ξ) = m Φ k (x)ψ k (ξ), we obtain k=1 H i r(ξ) = m k=1 F (Φ k U p i r )(ξ)ψ k (π X (ξ)). It immediately follows from (26) that a ji = a ij, i, j = 1,..., l, which shows that A is a Hermitian matrix. Further, for α 1,..., α l C we have l i,j=1 a ij α i α j = lim H r (ξ) 2 dξ 0, H r (ξ) = R n l Hr(ξ)α i i, which means that A is positive-definite. In the general case when g i C 0 ( S X ) one carries out the proof of (iii) by approximating the functions g i, i = 1,..., l in the uniform norm by finite sums of functions of the form Φ(x)ψ(ξ). Assertions (i) and (ii) are easy consequences of (iii). Indeed, setting l = 1, p 1 = p and g 1 = g, we obtain the relation µ pp, g 2 0, which holds for all g C 0 ( S X ), thus showing that µ pp is real and non-negative. To prove (ii) we represent an arbitrary function g = g(x, ξ) with compact support in the form g = g 1 g 2. Let l = 2, p 1 = p and p 2 = q. In view of (iii), µ pq, g = µ pq, g 1 g 2 = µ qp, g 2 g 1 = µ qp, g = µ qp, g and µ pq = µ qp. The proof is complete. We consider now a countable dense index subset D E. 12 i=1

13 Proposition 3 (cf. [15]). There exists a family of complex finite Borel measures µ pq x in S X with p, q D, x where is a subset of of full measure, such that µ pq = µ pq x dx, that is, for all Φ(x, ξ) C 0 ( S X ) the function x µ pq x (ξ), Φ(x, ξ) = Φ(x, ξ)dµ x pq (ξ) S X is Lebesgue-measurable on, bounded, and µ pq, Φ(x, ξ) = µ pq x (ξ), Φ(x, ξ) dx. Moreover, Var µ pq x 1 for all p, q D. Proof. We claim that pr µ pq meas for p, q E, where meas is the Lebesgue measure on. Assume first that p = q. By Lemma 4, the measure µ pp is non-negative. Next, in view of relation (25) with Φ 1 (x) = Φ 2 (x) = Φ(x) C 0 () and ψ(ξ) 1, µ pp, Φ(x) 2 = lim F (ΦUr p )(ξ)f (ΦUr p )(ξ)dξ = R n Ur p (x) 2 Φ(x) 2 dx Φ(x) 2 dx lim ( we use here Plancherel s equality and the estimate Ur p (x) 1 ). Thus, we see that that pr µ pp meas. Let p, q E, A be a bounded open subset of, and g = g(x, ξ) C 0 (A S X ), g 1. Let also g 1 = g/ g ( we set g 1 = 0 for g = 0 ) and g 2 = g. Then g 1, g 2 C 0 (A S X ), g = g 1 g 2, g 1 2 = g 2 2 = g and the matrix ( µ pp, g µ pq, g ) µ pq, g µ qq, g is positive-definite by Lemma 4; in particular, µ pq, g ( µ pp, g µ qq, g ) 1/2 (µ pp (A S X )µ qq (A S)) 1/2 meas(a). We take into account the inequalities pr µ pp meas and pr µ qq meas to obtain the last estimate. Since g can be an arbitrary function in C 0 (A S X ), g 1, we obtain the inequality µ pq (A S X ) meas(a). The measure µ pq is regular, therefore this estimate holds for all Borel subsets A of and It follows from (27) that for all ψ(ξ) C(S X ) we have pr µ pq meas. (27) pr (ψ(ξ)µ pq (x, ξ)) ψ pr µ pq ψ meas. (28) In view of (28) the measures pr (ψ(ξ)µ pq (x, ξ)) are absolutely continuous with respect to the Lebesgue measure, and the Radon-Nikodym theorem shows that pr (ψ(ξ)µ pq (x, ξ)) = h pq ψ (x) meas, 13

14 where the densities h pq ψ (x) are measurable on and, as seen from (28), h pq ψ (x) ψ. (29) We now choose a non-negative function K(x) C0 (R n ) with support in the unit ball such that K(x)dx = 1 and set K m (x) = m n K(mx) for m N. Clearly, the sequence of K m converges in D (R n ) to the Dirac δ-function ( that is, this sequence is an approximate unity ). Let B lim c m be a generalized Banach limit on the space l of bounded sequences c = {c m } m N, i.e. L(c) = B lim c m is a linear functional on l with the property: lim c m L(c) lim c m ( in particular for convergent sequences c = {c m } L(c) = lim c m ). For complex sequences c m = a m + ib m the Banach limits is defined by complexification: B lim c m = L(a) + il(b), where a = {a m }, b = {b m } are real and imaginary parts of the sequence c = {c m }, respectively. Modifying the densities h pq ψ (x) on subsets of measure zero, for instance, replacing them by the functions B lim ( obviously, the value h pq ψ ), we shall assume that for all x h pq ψ h pq ψ (y)k m(x y)dy (x) does not change for any Lebesgue point x of the function h pq ψ (x) = B lim h pq ψ (y)k m(x y)dy. (30) Let be the set of common Lebesgue points of the functions h pq ψ (x), u 0(x, p) = νx((p, 0 + )), and u 0 (x, p) = νx([p, 0 + )) = lim u 0(x, q), where p, q D and ψ belongs to F, some countable dense subset of C(S X ). The family of (p, q, ψ) is countable, q p therefore is of full measure. The dependence of h pq ψ on ψ, regarded as a map from C(S X) into L (), is clearly linear and continuous (in view of (29)), therefore it follows from the density of F in C(S X ) that x is a Lebesgue point of the functions h pq ψ (x) for all ψ(ξ) C(S X) and p, q D ( here we also take (30) into account ). For p, q D and x the equality l(ψ) = h pq ψ (x) defines a continuous linear functional in C(S X ); moreover, l 1 in view of (29). By the Riesz-Markov theorem this functional can be defined by integration with respect to some complex Borel measure µ pq x (ξ) in S X and Var µ pq x = l 1. Hence h pq ψ (x) = µpq x (ξ), ψ = ψ(ξ)dµ pq x (ξ) (31) S X for all ψ(ξ) C(S X ). 14

15 Equality (31) shows that the functions x S ψ(ξ)dµ pq x (ξ) are bounded and measurable for all ψ(ξ) C(S X ). Next, for Φ(x) C 0 () and ψ(ξ) C(S X ) we have ( ) Φ(x)ψ(ξ)dµ pq x (ξ) dx = Φ(x)h pq ψ (x)dx = S X Φ(x)dpr (ψ(ξ)µ pq ) = Φ(x)ψ(ξ)dµ pq (x, ξ). (32) S X Approximating an arbitrary function Φ(x, ξ) C 0 ( S X ) in the uniform norm by linear combinations of functions of the form Φ(x)ψ(ξ), we derive from (32) that the integral Φ(x, ξ)dµ pq x (ξ) is Lebesgue-measurable with respect to x, bounded, S X and ( ) Φ(x, ξ)dµ pq x (ξ) dx = Φ(x, ξ)dµ pq (x, ξ), S X S X that is, µ pq = µ pq x dx. Recall that Var µ pq x 1. The assumption that x are Lebesgue points of the functions u 0 (x, p), u 0 (x, p) for all p D will be used later. Observe that since p D E is a continuity point of the map p u 0 (x, p) in L 1 loc () then u 0 (x, p) = u 0 (x, p) a.e. in. By the construction x is a common Lebesgue point of the functions u 0 (x, p), u 0 (x, p), therefore ν 0 x({p}) = u 0 (x, p) u 0 (x, p) = 0 p D. (33) Remark 3. a) Since the H-measure is absolutely continuous with respect to x- variables identity (25) is satisfied for Φ 1 (x), Φ 2 (x) L 2 (). Indeed, by Proposition 3 we can rewrite this identity in the form: Φ 1 (x), Φ 2 (x) C 0 (), ψ(ξ) C(S X ) Φ 1 (x)φ 2 (x) ψ(ξ), µ pq x (ξ) dx = lim F (Φ 1 Ur p )(ξ)f (Φ 2 Ur q )(ξ)ψ(π X (ξ))dξ. (34) R n Both sides of this identity are continuous with respect to (Φ 1 (x), Φ 2 (x)) in L 2 () L 2 () and since C 0 () is dense in L 2 () we conclude that (34) is satisfied for each Φ 1 (x), Φ 2 (x) L 2 (); b) if x is a Lebesgue point of a function Φ(x) L 2 () then Φ(x) µ pq x, ψ(ξ) = lim lim F (ΦΦ m Ur p )(ξ)f (Φ m Ur q )(ξ)ψ(π X (ξ))dξ (35) R n for all ψ(ξ) C(S X ), where (ΦΦ m Ur p )(y) = Φ(y)Φ m (x y)ur p (y) and (Φ m Ur q )(y) = Φ m (x y)ur q (y). Indeed, it follows from (34) that lim R n F (ΦΦ m U p r )(ξ)f (Φ m U q r )(ξ)ψ(π X (ξ))dξ = 15 h pq ψ (y)φ(y)k m(x y)dy. (36)

16 Now, since x is a Lebesgue point of the functions h pq ψ (y) and Φ(y), and the function h pq ψ (y) is bounded, x is also a Lebesgue point for the product of these functions. Therefore, lim h pq ψ (y)φ(y)k m(x y)dy = Φ(x)h pq ψ (x) = Φ(x) µpq x, ψ(ξ), and (35) follows from (36) in the limit as m ; c) for x and each family p i D, ψ i (ξ) C(S X ), i = 1,..., l the matrix µ p ip j x, ψ i ψ j, i, j = 1,..., l is positive definite. Indeed, as follows from Lemma 4(iii), for α 1,..., α l C l µ p ip j x i,j=1, ψ i ψ j α i α j = lim i,j=1 l µ p ip j (y, ξ), Φ m (x y)ψ i (ξ)φ m (x y)ψ j (ξ) α i α j 0. Taking in the above property l = 2, p 1 = p, p 2 = q, ψ 1 (ξ) = ψ(ξ)/ ψ(ξ) ( ψ 1 = 0 for ψ = 0 ) and ψ 2 (ξ) = ψ(ξ), ( ψ(ξ) C(S X ), we ) obtain, as in the proof µ pp x, ψ µ pq x, ψ of Proposition 3, that the matrix µ pq x, ψ µ qq is positive definite. In x, ψ particular, µ pq x, ψ ( µ pp x, ψ µ qq x, ψ ) 1/2 and this easily implies that for any Borel set A S X µ pq x (A) (µ pp x (A)µ qq x (A)) 1/2. (37) We denote by θ(λ) the Heaviside function: { 1, λ > 0, θ(λ) = 0, λ 0. Below we shall frequently use the following simple estimate Lemma 5. Let p 0, p D, χ(λ) = θ(λ p 0 ) θ(λ p), V r (y) = νy(λ)), 0 Φ(y) L 2 (), x is a Lebesgue point of (Φ(y)) 2. Then χ(λ) d(ν r y(λ) + lim lim Φ m(x y)φ(y)v r (y) 2 2 Φ(x) u 0 (x, p 0 ) u 0 (x, p) 1/2 Proof. It is clear that V r (y) = u r (y, p 0 ) u r (y, p) + u 0 (y, p 0 ) u 0 (y, p) = sign(p p 0 )(u r (y, p 0 ) u r (y, p) + u 0 (y, p 0 ) u 0 (y, p)) 2 and, in particular, (V r (y)) 2 2V r (y). Therefore, 2 sign(p p 0 ) 0. p p0 Φ m (x y)φ(y)v r (y) 2 2 (Φ(y)) 2 K m (x y)(u r (y, p 0 ) u r (y, p) + u 0 (y, p 0 ) u 0 (y, p))dy. 16

17 Since p 0, p D E, u r (y, p 0 ) u r (y, p) u 0 (y, p 0 ) u 0 (y, p) as r weakly- in L () and we derive from the above inequality that lim Φ m(x y)φ(y)v r (y) sign(p p 0 ) (Φ(y)) 2 K m (x y)(u 0 (y, p 0 ) u 0 (y, p))dy. Now, passing to the limit as m and taking into account that x is a Lebesgue point of the bounded function u 0 (y, p 0 ) u 0 (y, p) as well as the function (Φ(y)) 2 ( therefore, x is a Lebesgue point of the product of these functions), we find lim lim Φ m(x y)φ(y)v r (y) 2 2 4(Φ(x)) 2 u 0 (x, p 0 ) u 0 (x, p). This implies the required relation lim lim Φ m(x y)φ(y)v r (y) 2 2 Φ(x) u 0 (x, p 0 ) u 0 (x, p) 1/2. To complete the proof it only remains to observe that, in view of (33), ν 0 x({p 0 }) = 0 and therefore u 0 (x, p) u 0 (x, p 0 ) as p p 0. The following statement is rather well-known. Lemma 6. Let U r (x) be a sequence bounded in L 2 (R n ) L 1 (R n ) and weakly convergent to zero, a(ξ) be a bounded function on R n such that a(ξ) 0 as ξ. Then a(ξ)f (U r )(ξ) 0 in L 2 (R n ). Proof. First, observe that by the assumption a(ξ) 0 at infinity for any ε > 0 we can choose R > 0 such that a(ξ) < ε for ξ > R. Then a(ξ) 2 F (U r )(ξ) 2 dξ ε 2 F (U r ) 2 = ε 2 U r 2 Cε 2, (38) ξ >R 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 a(ξ) 2 F (U r )(ξ) 2 dξ 0 (39) ξ R as r. It follows from (38), (39) that lim Since ε > 0 is arbitrary, we conclude that lim R n a(ξ) 2 F (U r )(ξ) 2 dξ Cε 2. R n a(ξ) 2 F (U r )(ξ) 2 dξ = 0, that is, a(ξ)f (U r )(ξ) 0 in L 2 (R n ). The proof is complete. 17

18 We now fix x, p 0, p D. Let L(p) R n be the smallest linear subspace containing supp µ pp 0 x, and L = L(p 0 ). As follows from (37), supp µ pp 0 x supp µ p 0p 0 x and therefore L(p) L. Suppose that f(y, λ) is a Caratheodory vector-function on R such that f(y, λ) L 2 loc (, C(R, Rn )), that is, M > 0 f(x, ) M, = max λ M f(x, λ) α M(x) L 2 loc(). (40) Since the space C(R, R n ) is separable with respect to the standard locally convex topology generated by seminorms M,, then, by the Pettis theorem (see [6], Chapter 3), the map x F (x) = f(x, ) C(R, R n ) is strongly measurable and in view of estimate (40) we see that F (x) L 2 loc (, C(R, Rn )), F (x) 2 L 1 loc (, C(R)). In particular (see [6], Chapter 3), the set f of common Lebesgue points of the maps F (x), F (x) 2 has full measure. For x f we have M > 0 lim K m (x y) F (x) F (y) M, dy = 0, Since, evidently, lim K m (x y) F (x) 2 F (y) 2 M, dy = 0. F (x) F (y) 2 M, 2 F (x) F (y) M, F (x) M, + F (x) 2 F (y) 2 M,, it follows from the above limit relations that for x f K m (x y) F (x) F (y) 2 M, dy = 0 M > 0. (41) lim Clearly, each x f is a Lebesgue point of all functions x f(x, λ), λ R. Let γx r = νx r νx. 0 Suppose that x f, p 0 D, χ(λ) = θ(λ p 1 ) θ(λ p 2 ), where p 1, p 2 D. For a vector-function h(y, λ) on R, which is Borel and locally bounded with respect to the second variable, we denote I r (h)(y) = h(y, λ)dγy(λ). r In view of the strong measurability of F (x) and (40) we see that I r = I r (f χ)(y) L 2 loc () ( see Remark 1 ). We also denote by L, L the spaces obtained by orthogonal projections of L on the subspaces X, X, respectively: L = P1 (L), L = P 2 (L). Proposition 4. Assume that f(x, λ) L, and ρ(ξ) C (R n ) is a function such that 0 ρ(ξ) 1 and ρ(ξ) = 0 for ξ 2 + ξ 4 1, ρ(ξ) = 1 for ξ 2 + ξ 4 2. Then ψ(ξ) C(S X ) lim lim ρ(ξ)ξ F (Φ m I r (f χ))(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = 0. Here Φ m = Φ m (x y) = K m (x y) and I r (f χ) are supposed to be functions of the variable y. 18

19 Proof. Note that I r (y) f(y, λ) χ(λ) d γ r y (λ) 2α M (y), (42) where M = sup ν r x. Let us first show that for each m N ρ(ξ) ξ F (Φ m I r )(ξ) lim F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = 0. (43) For that, it is sufficient to demonstrate that ρ(ξ) ξ ( ξ 2 + ξ 4 ) 1/2 F (Φ mi r )(ξ) 0 in L 2 (R n ). (44) Remark that the sequence Φ m I r (y), r N is bounded in L 2 (R n ) and in L 1 (R n ) ( since supp Φ m is compact ) and weakly converges to zero ( in view of the weak convergence νx r νx 0 ). Hence, (44) follows from Lemma 6. We only need to demonstrate that the function ρ(ξ) ξ a(ξ) = ( ξ 2 + ξ 4 ) 1/2 satisfies the assumptions of this Lemma. First, we show that a(ξ) 1. Indeed, for ξ 2 + ξ 4 1 the value ρ(ξ) = 0 while in the case ξ 2 + ξ 4 > 1 we have ρ(ξ) ξ ( ξ 2 + ξ 4 ) 1/2 min( ξ, 1/ ξ ) 1. Then, observe that for ξ 2 + ξ 4 R 4 > 0 a(ξ) ξ ( ξ 2 + ξ 4 ) 1/2 ( ξ 2 + ξ 4 ) 1/4 R 1. Therefore, a(ξ) 0 as ξ. Thus, assumptions of Lemma 6 are satisfied and by Lemma 6 we conclude that (44), (43) hold. In view of (43), lim lim ρ(ξ)ξ F (Φ m I r )(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = lim lim ρ(ξ) ξ F (Φ m I r )(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ. (45) Let g(λ) = f(x, λ), I r = I r (gχ)(y) = g(λ)χ(λ)dγ r y(λ), M = sup ν r y. Then I r I r f(y, λ) f(x, λ) d γ r y (λ) 2 F (y) F (x) M,. 19

20 This and the Plancherel identity imply that ρ(ξ) ξ F (Φ m (I r I r))(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ ψ F (Φ m (I r I r)) 2 F (Φ m U p 0 r ) 2 ψ Φ m (I r I r) 2 2 ψ (R n K m (x y) F (y) F (x) 2 M, ) 1/2. It follows from the above estimate and (41) that lim lim ρ(ξ) ξ F (Φ m I r )(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ ρ(ξ) ξ F (Φ m I r)(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ lim lim ρ(ξ) ξ F (Φ m (I r I r))(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = 0 and, in view of this relation and (45), it is sufficient to prove that lim lim ρ(ξ) ξ F (Φ m I r)(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = 0. (46) The vector-function g(λ) is continuous and does not depend on y. Therefore for any ε > 0 there exists a vector-valued function h(λ) of the form h(λ) = where v i L and p i D such that g χ h ε on R. Using again the Plancherel s identity and the fact that I r I r (h) = (g χ h)(λ)dγy(λ) r (g χ h)(λ) d γy (λ) r 2ε, k i=1 v i θ(λ p i ), we obtain ρ(ξ) ξ F (Φ m I r)(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ ρ(ξ) ξ F (Φ m I r (h))(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ ψ Φ m I r (g χ h) 2 2ε ψ Φ m 2 = 2ε ψ. (47) Since ( ) k I r (h)(y) = v i θ(λ p i ) dγy(λ) r = i=1 20 k i=1 v i U p i r (y),

21 it follows from (35) the limit relation lim lim ρ(ξ) ξ F (Φ m I r (h))(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = k i=1 µ p ip 0 x, (v i ξ)ψ(ξ). (48) Here we also take Remark 2 into account. Since ρ(ξ)ψ(π X (ξ)) = ψ(π X (ξ)) for large ξ then, by this Remark, for i = 1,..., k ρ(ξ) ξ v i F (Φ m Ur pi )(ξ) lim F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = lim R n ξ vi F (Φ m U pi r )(ξ) ( ξ 2 + ξ 4 ) 1/2 F (Φ m U p 0 r )(ξ)ψ(π X (ξ))dξ = µ p ip 0 (y, ξ), K m (x y)(v i ξ)ψ(ξ). Now observe that supp µ p ip 0 x L(p 0 ) = L, and for each ξ L v i ξ = 0 because ξ L k while v i L. Hence µ p ip 0 x, (v i ξ)ψ(ξ) = 0, and it follows from (48) that i=1 lim lim ρ(ξ) ξ F (Φ m I r (h))(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = 0. This relation together with (47) yields lim lim ρ(ξ) ξ F (Φ m I r)(ξ) F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ 2ε ψ, and since ε > 0 is arbitrary we claim that (46) holds. This completes the proof. Let Q(x, λ) be a Caratheodory matrix-valued function, which ranges in the space Sym n of symmetric matrices of order n such that Q(x, λ) L 2 loc (, C(R, Sym n)). Denote Q the set of full measure consisting of common Lebesgue points of the maps x G(x) = Q(x, ) C(R, Sym n ), x G(x) 2 C(R). As can be easily verified, for x Q the following relation similar to (41) holds lim K m (x y) G(x) G(y) 2 M, dy = 0 M > 0. (49) Let x Q, p 0, p 1, p 2 D, χ(λ) = θ(λ p 1 ) θ(λ p 2 ), and let J r (y) = J r (Q)(y) = χ(λ)q(y, λ)dγy(λ), r ρ(ξ) be a function as in Proposition 4. Also assume that for L = L(p 0 ) Q(x, λ)ξ ξ = 0 ξ L = P 2 (L) 21

22 (recall that P 2 is the orthogonal projection onto X ). Proposition 5. Under the above notations for each ψ(ξ) C(S X ) lim lim ρ(ξ)f (Φ m J r )(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 2 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = 0. (50) Proof. Denote Q(λ) = Q(x, λ) ( here x is the fixed above point ), Jr (y) = χ(λ) Q(λ)dγ r y (λ). Then J r J r Q(y, λ) Q(x, λ) d γ r y (λ) 2 G(y) G(x) M, where M = sup νx r. This and the Plancherel identity imply that ρ(ξ)f (Φ m (J r J r ))(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ ψ F (Φ m (J r J r )) 2 F (Φ m U p 0 r ) 2 ψ Φ m (J r J r ) 2 2 ψ (R n K m (x y) G(y) G(x) 2 M, ) 1/2. It follows from the above estimate and (49) that lim lim ρ(ξ)f (Φ m J r )(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ ρ(ξ)f (Φ m Jr )(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ lim lim ρ(ξ)f (Φ m (J r J r ))(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = 0 and, in view of this relation we have to prove that lim lim ρ(ξ)f (Φ m Jr )(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = 0. (51) Introduce the linear space Y of symmetric matrices A, satisfying the property Aξ ξ = 0 for ξ L. Since the matrix-valued function Q(λ) ranges in Y and does not depend on k y for every ε > 0 one can find a step function H(λ) = θ(λ p i )Q i, where p i D, Q i Y for each i = 1,..., k such that χ(λ) Q(λ) H(λ) < ε for all λ R. We denote J r(y) = H(λ)dγ r y(λ) and observe that i=1 J r (y) J r(y) J r(y) = k i=1 U p i r (y)q i, (52) Q(λ) H(λ) χ(λ) d γ r y (λ) 2ε. (53) 22

23 We also remark that F (Φ m ( J r J r))(ξ) ξ ξ ( ξ 2 + ξ 4 ) 1/2 F (Φ m( J r J r))(ξ) ξ 2 /( ξ 2 + ξ 4 ) 1/2 F (Φ m ( J r J r))(ξ). The latter estimate and (53) imply that ρ(ξ)f (Φ m Jr )(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ ρ(ξ)f (Φ m J r)(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = ρ(ξ)f (Φ m ( J r J r))(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ ψ F (Φ m ( J r J r)) 2 F (Φ m U p 0 r ) 2 = ψ Φ m ( J r J r) 2 Φ m U p 0 r 2 1/2 ψ Φ m ( J r J r) 2 = ψ K m (x y) (R J r (y) J r(y) dy) 2 2ε ψ. (54) n We also use that U p 0 r 1 and therefore Φ m U p 0 r 2 1. In view of (52) ρ(ξ)f (Φ m J r)(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = k ρ(ξ)f (Φ m Ur pi )(ξ)q i ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ, i=1 and by relation (35) and Remark 2 we find lim lim ρ(ξ)f (Φ m J r)(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ = k i=1 µ p ip 0 x ψ(ξ)q i ξ ξ = 0 (55) because supp µ p ip 0 x L and therefore Q i ξ ξ = 0 on supp µ p i p 0 x ( recall that Q i ξ ξ = 0 for ξ L ). By (54), (55) we obtain the relation lim lim ρ(ξ)f (Φ m Jr )(ξ) ξ ξ F (Φ m U p 0 R n ( ξ 2 + ξ 4 ) 1/2 r )(ξ)ψ(π X (ξ))dξ 2ε ψ and since ε > 0 is arbitrary we conclude that (51) holds. The proof is complete. In the sequel we will need the following simple result. 23

24 Lemma 7. Let { ξ k k = 1,..., l } L be a basis in L. Then there exists a positive constant C such that for every v R n, Q Sym n v 1 + Q 1 C max k=1,...,l iv ξ k + Q ξ k ξ k, where v 1 = P v, Q 1 = P Q P, P, P are orthogonal projections on the spaces L, L, respectively, and i = 1. Proof. We introduce the linear spaces S = { Q Sym n Q = P Q P }, H = L S and remark that p(v, Q) = max iv ξ k + Q ξ k ξ k is a norm in H. Indeed, it is clear k=1,...,l that p is a seminorm. To prove that p is a norm, suppose that p(v, Q) = 0. Then v ξ k = Q ξ k ξ k = 0 and since vectors ξ k, ξ k generate spaces L, L, respectively, we claim that v ξ = 0 for all ξ L and Qξ ξ = 0 for all ξ L. Since v L we see that v = 0. Further, since Q S we find that for every ξ R n Qξ ξ = P Q P ξ ξ = Q P ξ P ξ = 0, and we conclude that Q = 0. It is well-known that any two norms in finite-dimensional space are equivalent. Applying this property to the norms p(v, Q) and p 1 (v, Q) = v + Q and using the relations v ξ k = v 1 ξ k, Q ξ k ξ k = Q P ξ k P ξ k = Q 1 ξk ξ k, k = 1,... l, we find that for some constant C > 0 v 1 + Q 1 C max k=1,...,l iv 1 ξ k + Q 1 ξk ξ k = C max k=1,...,l iv ξ k + Q ξ k ξ k, as was to be proved. Corollary 1. There exist functions ψ k (ξ) C(S X ), k = 1,..., l = dim L and a constant C > 0 such that, in the notations of Lemma 7, for all v R n, Q Sym n such that Q 0 v 1 + Q 1 C max k=1,...,l µp 0p 0 x, (iv ξ + Q ξ ξ)ψ k (ξ). (56) Proof. Remark that the measure µ p 0p 0 x 0. If µ p 0p 0 x = 0 then the both parts of equality (56) equal zero, and this equality is evidently satisfied. Thus, suppose that µ p 0p 0 x (S X ) > 0. Since L is a linear span of supp µ p 0p 0 x, we can choose functions ψ k (ξ) C(S X ), k = 1,..., l such that ψ k (ξ) 0, ψ k (ξ)dµ p 0p 0 x = 1 for all k = 1,..., l, and the family ξ k = ξψ k (ξ)dµ p 0p 0 x, k = 1,..., l is a basis in L. By Lemma 7 there exists a constant C > 0 such that for all v R n, Q Sym n v 1 + Q 1 C max k=1,...,l iv ξ k + Q ξ k ξ k, (57) where v 1 = P v, Q 1 = P Q P. Now, we observe that ξ k = ξψ k (ξ)dµ p 0p 0 x (ξ), ξk = 24 ξψ k (ξ)dµ p 0p 0 x (ξ).

25 Therefore, v ξ k = v ξψ k (ξ)dµ p 0p 0 x (ξ), and if Q 0 then Q ξ k ξ k = Q ξψ k (ξ)dµ p 0p 0 x (ξ) ξψ k (ξ)dµ p 0p 0 x (ξ) Q ξ ξψ k (ξ)dµ p 0p 0 x (ξ) by Jensen s inequality applied to the convex function ξ Q ξ ξ. In view of the above relation, (56) readily follows from (57) ( we also take into account that for real a the function f(x) = ia + x increases on [0, + ) ). The proof is complete. 4 Localization principle and strong precompactness of bounded sequences of measurevalued functions In this Section we need 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 pseudodifferential 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 Ṙ n = (R \ {0}) n = { ξ = (ξ 1,..., ξ n ) n ξ k 0 }. The following statement readily follows from the known Marcinkiewicz multiplier theorem (see [19][Chapter 4]). Theorem 3. Suppose that a(ξ) C n (Ṙn ) be a function such that for some constant C ξ α D α a(ξ) C ξ Ṙn (58) for every multi-index α = (α 1,..., α n ) such that α = α α n n. Then a(ξ) M d for all d > 1. Here we use the standard notations ξ α = n ( ) αi n i=1 (ξ i) α i, D α =. Actually ξ i=1 i (see [19]), it is sufficient to require that (58) is satisfied for multi-indexes α such that α i {0, 1}, i = 1,..., n. We also need the following simple lemma. Lemma 8. Let h(y, z) C n ((R l R n l )\{0}) be such that for some k N, γ R k=1 t > 0 h(t k y, tz) = t γ h(y, z). (59) 25

26 Then there exists a constant C > 0 such that for each multi-indexes α = (α 1,..., α l ), β = (β 1,..., β n l ), α + β n and all y R l z R n l, 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 (59) for all t > 0 Taking t = ( y 2 + z 2k ) 1 2k 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). in this relation, we arrive at 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 ), (60) 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, and relation (60) 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) for all y, z 0. The proof is complete. C( y 2 + z 2k ) γ 2k y α z β 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 π X : R n S X be the projection defined in Section 2. 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 with the properties indicated in Proposition 4, namely: 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 [2][Chapter 6] ) then we can assume that X = R l = {ξ R n ξ = (y 1,..., y l, 0,..., 0) } while X = {ξ R n ξ = (0,..., 0, z 1,..., z n l ) }. 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)) satisfies the assumptions of Lemma 8 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 (58) of Theorem 3 is satisfied. By this Theorem we conclude that a 1 (ξ) M d for each d > 1. 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 8 with y replaced by (s, y) R l+1, and k = γ = 2. By this Lemma 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. 26