The Schwartz kernel theorem and distributions of finite order Nenad Antoni´c Department of Mathematics Faculty of Science University of Zagreb A conference in memory of Todor Gramchev Torino, 1st February 2017 Joint work with Marko Erceg and Marin Miˇsur Distributions of anisotropic order H-distributions Functions of anisotropic smoothness Definition and tensor products Conjectures Schwartz kernel theorem Statement and strategies The proof Consequence for H-distributions 2 Theorem. [H¨ormander-Mihlin] Let 2 L1(Rd) have partial derivatives of d order less than or equal to κ = [ 2 ] + 1. If for some k > 0 Z d α 2 2 d−2jαj (8r > 0)(8α 2 N0) jαj 6 κ =) j@ (ξ)j dξ 6 k r ; r 2 6jξj6r then for any p 2 h1; 1i and the associated multiplier operator A there exists a Cd (depending only on the dimension d) such that 1 kA k p p C max p; (k + k k ) : L !L 6 d p − 1 1 κ d−1 d For 2 C (S ), extended by homogeneity to R∗, we can take k = k kCκ . H¨ormander-Mihlintheorem : Rd ! C is a Fourier multiplier on Lp(Rd) if F¯( F(θ)) 2 Lp(Rd) ; for θ 2 S(Rd); and S(Rd) 3 θ 7! F¯( F(θ)) 2 Lp(Rd) p d p d can be extended to a continuous mapping A :L (R ) ! L (R ). 3 κ d−1 d For 2 C (S ), extended by homogeneity to R∗, we can take k = k kCκ . H¨ormander-Mihlintheorem : Rd ! C is a Fourier multiplier on Lp(Rd) if F¯( F(θ)) 2 Lp(Rd) ; for θ 2 S(Rd); and S(Rd) 3 θ 7! F¯( F(θ)) 2 Lp(Rd) p d p d can be extended to a continuous mapping A :L (R ) ! L (R ). Theorem. [H¨ormander-Mihlin] Let 2 L1(Rd) have partial derivatives of d order less than or equal to κ = [ 2 ] + 1. If for some k > 0 Z d α 2 2 d−2jαj (8r > 0)(8α 2 N0) jαj 6 κ =) j@ (ξ)j dξ 6 k r ; r 2 6jξj6r then for any p 2 h1; 1i and the associated multiplier operator A there exists a Cd (depending only on the dimension d) such that 1 kA k p p C max p; (k + k k ) : L !L 6 d p − 1 1 3 H¨ormander-Mihlintheorem : Rd ! C is a Fourier multiplier on Lp(Rd) if F¯( F(θ)) 2 Lp(Rd) ; for θ 2 S(Rd); and S(Rd) 3 θ 7! F¯( F(θ)) 2 Lp(Rd) p d p d can be extended to a continuous mapping A :L (R ) ! L (R ). Theorem. [H¨ormander-Mihlin] Let 2 L1(Rd) have partial derivatives of d order less than or equal to κ = [ 2 ] + 1. If for some k > 0 Z d α 2 2 d−2jαj (8r > 0)(8α 2 N0) jαj 6 κ =) j@ (ξ)j dξ 6 k r ; r 2 6jξj6r then for any p 2 h1; 1i and the associated multiplier operator A there exists a Cd (depending only on the dimension d) such that 1 kA k p p C max p; (k + k k ) : L !L 6 d p − 1 1 κ d−1 d For 2 C (S ), extended by homogeneity to R∗, we can take k = k kCκ . 3 µ is the H-distribution corresponding to (a subsequence of) (un) and (vn). d d If (un), (vn) are defined on Ω ⊆ R , extension by zero to R preserves the convergence, and we can apply the Theorem. µ is supported on Cl Ω × Sd−1. p d q d 0 We distinguish un 2 L (R ) and vn 2 L (R ). For p > 2, p 6 2 and we can 2 take q > 2; this covers the L case (including un = vn). 2 d The assumptions imply un; vn −* 0 in Lloc(R ), resulting in a distribution µ of order zero (an unbounded Radon measure, not a general distribution). The novelty in Theorem is for p < 2. p d k q d l For vector-valued un 2 L (R ; C ) and vn 2 L (R ; C ), the result is a matrix valued distribution µ = [µij ], i 2 1::k and j 2 1::l. The H-distribution would correspond to a non-diagonal block for an H-measure. Existence of H-distributions p d Theorem. [N.A. & D. Mitrovi´c(2011)] If un −* 0 in L (R ) and ∗ q d 0 vn −−* v in L (R ) for some q > maxfp ; 2g, then there exist subsequences 0 d d−1 (un0 ), (vn0 ) and a complex valued distribution µ 2 D (R × S ), such that 1 d κ d−1 for every '1;'2 2 Cc (R ) and 2 C (S ) we have: Z Z lim A ('1un0 )(x)('2vn0 )(x)dx = lim ('1un0 )(x)A ('2vn0 )(x)dx 0 0 n Rd n Rd = hµ, '1'2 i : 4 d d If (un), (vn) are defined on Ω ⊆ R , extension by zero to R preserves the convergence, and we can apply the Theorem. µ is supported on Cl Ω × Sd−1. p d q d 0 We distinguish un 2 L (R ) and vn 2 L (R ). For p > 2, p 6 2 and we can 2 take q > 2; this covers the L case (including un = vn). 2 d The assumptions imply un; vn −* 0 in Lloc(R ), resulting in a distribution µ of order zero (an unbounded Radon measure, not a general distribution). The novelty in Theorem is for p < 2. p d k q d l For vector-valued un 2 L (R ; C ) and vn 2 L (R ; C ), the result is a matrix valued distribution µ = [µij ], i 2 1::k and j 2 1::l. The H-distribution would correspond to a non-diagonal block for an H-measure. Existence of H-distributions p d Theorem. [N.A. & D. Mitrovi´c(2011)] If un −* 0 in L (R ) and ∗ q d 0 vn −−* v in L (R ) for some q > maxfp ; 2g, then there exist subsequences 0 d d−1 (un0 ), (vn0 ) and a complex valued distribution µ 2 D (R × S ), such that 1 d κ d−1 for every '1;'2 2 Cc (R ) and 2 C (S ) we have: Z Z lim A ('1un0 )(x)('2vn0 )(x)dx = lim ('1un0 )(x)A ('2vn0 )(x)dx 0 0 n Rd n Rd = hµ, '1'2 i : µ is the H-distribution corresponding to (a subsequence of) (un) and (vn). 4 p d q d 0 We distinguish un 2 L (R ) and vn 2 L (R ). For p > 2, p 6 2 and we can 2 take q > 2; this covers the L case (including un = vn). 2 d The assumptions imply un; vn −* 0 in Lloc(R ), resulting in a distribution µ of order zero (an unbounded Radon measure, not a general distribution). The novelty in Theorem is for p < 2. p d k q d l For vector-valued un 2 L (R ; C ) and vn 2 L (R ; C ), the result is a matrix valued distribution µ = [µij ], i 2 1::k and j 2 1::l. The H-distribution would correspond to a non-diagonal block for an H-measure. Existence of H-distributions p d Theorem. [N.A. & D. Mitrovi´c(2011)] If un −* 0 in L (R ) and ∗ q d 0 vn −−* v in L (R ) for some q > maxfp ; 2g, then there exist subsequences 0 d d−1 (un0 ), (vn0 ) and a complex valued distribution µ 2 D (R × S ), such that 1 d κ d−1 for every '1;'2 2 Cc (R ) and 2 C (S ) we have: Z Z lim A ('1un0 )(x)('2vn0 )(x)dx = lim ('1un0 )(x)A ('2vn0 )(x)dx 0 0 n Rd n Rd = hµ, '1'2 i : µ is the H-distribution corresponding to (a subsequence of) (un) and (vn). d d If (un), (vn) are defined on Ω ⊆ R , extension by zero to R preserves the convergence, and we can apply the Theorem. µ is supported on Cl Ω × Sd−1. 4 p d k q d l For vector-valued un 2 L (R ; C ) and vn 2 L (R ; C ), the result is a matrix valued distribution µ = [µij ], i 2 1::k and j 2 1::l. The H-distribution would correspond to a non-diagonal block for an H-measure. Existence of H-distributions p d Theorem. [N.A. & D. Mitrovi´c(2011)] If un −* 0 in L (R ) and ∗ q d 0 vn −−* v in L (R ) for some q > maxfp ; 2g, then there exist subsequences 0 d d−1 (un0 ), (vn0 ) and a complex valued distribution µ 2 D (R × S ), such that 1 d κ d−1 for every '1;'2 2 Cc (R ) and 2 C (S ) we have: Z Z lim A ('1un0 )(x)('2vn0 )(x)dx = lim ('1un0 )(x)A ('2vn0 )(x)dx 0 0 n Rd n Rd = hµ, '1'2 i : µ is the H-distribution corresponding to (a subsequence of) (un) and (vn). d d If (un), (vn) are defined on Ω ⊆ R , extension by zero to R preserves the convergence, and we can apply the Theorem. µ is supported on Cl Ω × Sd−1. p d q d 0 We distinguish un 2 L (R ) and vn 2 L (R ). For p > 2, p 6 2 and we can 2 take q > 2; this covers the L case (including un = vn).