Anisotropic distributions and applications Marin Miˇsur email: [email protected] University of Zagreb joint work with Nenad Antoni´c(UniZg) and Marko Erceg (SISSA) August 11, 2017 H-measures12 2 r d+1 Theorem 1. If (un)n2N is a sequence in Lloc(Ω; R ), Ω ⊂ R , such that 2 un * 0 in Lloc(Ω), then there exists subsequence (un0 )n0 ⊂ (un)n and positive jk d+1 d complex bounded measure µ = fµ gj;k=1;:::;r on R × S such that for all d '1;'2 2 C0(Ω) and 2 C(S ), Z j k jk lim ('1u 0 )(ξ)A ξ ('2u 0 )(ξ)dx = hµ ;'1'¯2 i 0 n n n !1 Ω jξj Z jk = '1(x)'2(x) (ξ)dµ (x; ξ) Rd+1×Sd where A ξ is the multiplier operator with the symbol (ξ=jξj). jξj 1L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh 115A (1990) 193{230. 2P. G´erard, Microlocal defect measures, Comm. Partial Diff. Eq. 16 (1991) 1761{1794. H-distributions5 p d ∗ q d Theorem 2. If un −* 0 in Lloc(R ) and vn −* v in Lloc(R ) for some 0 p 2 h1; 1i and q ≥ p , then there exist subsequences (un0 ), (vn0 ) and a complex valued distribution µ 2 D0(Rd × Sd−1), such that, for every d κ d−1 '1;'2 2 Cc(R ) and 2 C (S ), for κ = [d=2] + 1, one has: Z Z lim A ('1un0 )(x)('2vn0 )(x)dx = lim ('1un0 )(x)A ('2vn0 )(x)dx 0 0 n !1 Rd n !1 Rd = hµ, '1'2 i; p d p d where A :L (R ) −! L (R ) is the Fourier multiplier operator with symbol 2 Cκ(Sd−1). Existing applications are related to the velocity averaging3 and Lp − Lq compactness by compensation4. 3M. Lazar, D. Mitrovi´c, On an extension of a bilinear functional on Lp(Rd) × E to Bochner spaces with an application to velocity averaging, C. R. Math. Acad. Sci. paris 351 (2013) 261{264. 4M. Miˇsur,D. Mitrovi´c, On a generalization of compensated compactness in the Lp − Lq setting, Journal of Functional Analysis 268 (2015) 1904{1927. 5N. Antoni´c,D. Mitrovi´c, H-distributions: An Extension of H-Measures to an Lp − Lq Setting, Abs. Appl. Analysis Volume 2011, Article ID 901084, 12 pages. Introduction Anisitropic distributions The Schwartz kernel theorem Peetre's result Anisotropic distributions Let X and Y be open sets in Rd and Rr (or C1 manifolds of dimenions d and r) and Ω ⊆ X × Y an open set. By Cl;m(Ω) we denote the space of functions d r f on Ω, such that for any α 2 N0 and β 2 N0, if jαj ≤ l and jβj ≤ m, α;β α β @ f = @x @y f 2 C(Ω). Cl;m(Ω) becomes a Fr´echetspace if we define a sequence of seminorms l;m α;β 1 pK (f) := max k@ fkL (Kn) ; n jα|≤l;jβ|≤m where Kn ⊆ Ω are compacts, such that Ω = [n2NKn and Kn ⊆ IntKn+1, Consider the space l;m [ l;m C (Ω) := C (Ω) ; c Kn n2N and equip it by the topology of strict inductive limit. Anisotropic distributions II Definition. A distribution of order l in x and order m in y is any linear l;m functional on Cc (Ω), continuous in the strict inductive limit topology. We 0 denote the space of such functionals by Dl;m(Ω). Conjecture. Let X; Y be C1 manifolds and let u be a linear functional on l;m 0 Cc (X × Y ). If u 2 D (X × Y ) and satisfies 1 1 (8K 2 K(X))(8L 2 K(Y )(9C > 0)(8' 2 CK (X))(8 2 CL (Y )) l m jhu; ' ij ≤ CpK (')pL ( ); l;m then u can be uniquely extended to a continuous functional on Cc (X × Y ) 0 (i.e. it can be considered as an element of Dl;m(X × Y )). The Schwartz kernel theorem6 Let X and Y be two C1 manifolds. Then the following statements hold: Theorem 3. a) Let K 2 D0(X × Y ). Then for every ' 2 D(X), the linear form K' defined as 7! hK; ' i is a distribution on Y . Furthermore, the 0 mapping ' 7! K', taking D(X) to D (Y ) is linear and continuous. b) Let A : D(X) !D0(Y ) be a continous linear operator. Then there exists unique distribution K 2 D0(X × Y ) such that for any ' 2 D(X) and 2 D(Y ) hK; ' i = hK'; i = hA'; i: 6J. Dieudonn´e, El´ementsd'Analyse,´ Tome VII, Editions´ Jacques Gabay, 2007. The Schwartz kernel theorem for anisotropic distributions Let X and Y be two C1 manifolds of dimensions d and r, respectively. Then the following statements hold: 0 l Theorem 4. a) Let K 2 Dl;m(X × Y ). Then for every ' 2 Cc(X), the linear form K' defined as 7! hK; ' i is a distribution of order not more than m l 0 on Y . Furthermore, the mapping ' 7! K', taking Cc(X) to Dm(Y ) is linear and continuous. l 0 b) Let A :Cc(X) !Dm(Y ) be a continous linear operator. Then there exists unique distribution K 2 D0(X × Y ) such that for any ' 2 D(X) and 2 D(Y ) hK; ' i = hK'; i = hA'; i: 0 Furthermore, K 2 Dl;r(m+2)(X × Y ). How to prove it? Standard attempts: ◦ regularisation? (Schwartz) ◦ constructive proof? (Simanca, Gask, Ehrenpreis) ◦ nuclear spaces? (Treves) Use the structure theorem of distributions (Dieudonne). There are two steps: Step I: assume the range of A is C(Y ) Step II: use structure theorem and go back to Step I Consequence: H-distributions are of order 0 in x and of finite order not greater than d(κ + 2) with respect to ξ. A variant by Bogdanovicz's result7 We can reformulate the main result of Bogdanowicz's article to our setting: 1 l Theorem 5. For every bilinear functional B on the space Cc (X1) × Cc(X2) which is continuous with respect to each variable separately, there exists a 0 unique anisotropic distribution T 2 D1;l(X1 × X2) such that 1 l B('; φ) = hT;' ⊗ φi ;' 2 Cc (X1); φ 2 Cc(X2) : It is worth noting that Bogdanowicz's result also holds when X2 is a smooth manifold and that only elementary properties of Frechet and (LF)-spaces were used to prove it. The same result can be obtained using the adjoint of operator A. 7W. M. Bogdanowicz: A proof of Schwartz's theorem on kernels, Studia Math. 20 (1961) 77{85. Additional results8 0 Lema 1. If u 2 Dl;m(X1 × X2) is of compact support such that 1 supp u ⊂ f0g × X2, then for any Φ 2 Cc (X1 × X2) it holds: X u = huα; Φαi ; d α2N0 ;jα|≤l 0 α where uα 2 Dm(X2) and Φα(y) = Dx Φ(0; y). 0 Corollary 1. If u 2 Dl;m(X1 × X2) has compact support such that supp u ⊂ fx0g × X2, for some x0 2 X1, then X α u = Dx δx0 ⊗ uα ; d α2N0 ;jα|≤l 0 where uα 2 Dm(X2). 1 0 Theorem 6. Let A :Cc (X) !Dm(X) be a continuous map. Its kernel is 1 supported by the diagonal f(x; x): x 2 Xg if and only if for any ' 2 Cc (X): X α A' = aα ⊗ D '; d α2N0 0 where aα 2 Dm(X) and the above sum is locally finite. Moreover, this representation is unique. 8L. H¨ormander: The Analysis of Linear Partial Differential Operators I, Springer, 1990. Peetre's results9 1 1 Theorem 7. Let A :Cc (Ω) ! Cc (Ω) be a linear mapping such that the following holds: 1 supp(Au) ⊂ supp(u) ; u 2 Cc (Ω): Then A is a differential operator on Ω with C1 coefficients. 1 0 Theorem 8. Let A :Cc (Ω) !D (Ω) be linear operator such that 1 supp (Af) ⊂ supp f for f 2 Cc (Ω): Then there exists locally finite family of 0 distributions (aα) 2 D (Ω), unique on Ω n Λ, such that it holds: X α 1 supp Af − aαD f ⊂ Λ ; f 2 Cc (Ω) : α 9J. Peetre: Une caracterisation abstraite des operateurs differentiels, Math. Scand. 7 (1959) 211{218; Rectification, ibid 8 (1960) 116{120. Peetre's result with distributions of finite order 1 0 Theorem 9. Let A :Cc (Ω) !Dm(Ω) be linear operator such that 1 supp (Af) ⊂ supp f ; f 2 Cc (Ω) : (1) 0 Then there exists locally finite family of distributions (aα) 2 Dm(Ω), unique on Ω n Λ, such that it holds: X α 1 supp Af − aαD f ⊂ Λ ; f 2 Cc (Ω) : α Idea of the proof Proof. Let U ⊂ Ω be an open and relatively compact set. Then there exists j = j(U) 2 N such that for any x0 2 U n Λ, there is a neighbourhood V of x0 such that 1 m jhAf; gij ≤ Ckfkj kgkm ; f 2 Cc (V ); g 2 Cc (V ) : 0 Schwartz kernel theorem gives existence of K 2 D1;m(V × V ) such that hAf; gi = hK; f ⊗ gi. The locality assumption implies that the distribution K is supported on a diagonal of a set V : supp K ⊂ f(x; x): x 2 V g : The theorem on the diagonal support gives: X α 1 A' = aα ⊗ D ';' 2 Cc (V ) ; d α2N0 0 where family (aα) ⊂ Dm(V ) is locally finite.