A Note on Mixed Norm Spaces

Size: px

Start display at page:

Download "A Note on Mixed Norm Spaces"

Owen Walters
5 years ago
Views:

1 1/21 A Note on Mixed Norm Spaces Nadia Clavero University of Barcelona Seminari SIMBa April 28, 2014

2 2/21 1 Introduction 2 Sobolev embeddings in rearrangement-invariant Banach spaces 3 Sobolev embeddings in mixed norm spaces 4 Critical case of the classical Sobolev embedding

3 Introduction 3/21 Rearrangement-invariant Banach function spaces Given a finite interval I, we denote I n = Definition n {}}{ I... I, n N. A rearrangement invariant Banach function space (briefly an r.i. space) is defined as X (I n ) = { f M(I n ) : f X (I n ) < }, where X (I n ) satisfies certain properties. Examples The Lebesgue spaces L p (I n ), where ( f L p (I n ) = I f (x) p dx ) 1/p, 1 p < ; n inf { C 0 : f (x) C a.e }, p =.

4 Introduction 4/21 Mixed norm spaces Let n N, n 2 and k {1,..., n}. For any x I n, we denote x k = (x 1,..., x k 1, x k+1,..., x n ) I n 1. Definition The Benedek-Panzone spaces are defined as R k (X, Y ) = { f M(I n ) : f Rk (X,Y ) < }, where f Rk (X,Y ) = ψ k (f, Y ) X (I n 1 ), ψ k(f, Y )( x k ) = f ( x k, ) Y (I).

5 Introduction 5/21 Mixed norm spaces Examples: The Lebesgue spaces L 1 (I) = R 1 (L 1, L 1 ); f R1(L 1,L 1 ) I = ψ 1 (f, L 1 )( x 1 )d x 1 = f ( x 1, x 1 ) dx 1 d x 1. n 1 I n 1 I The Benedek-Panzone spaces R n (L 1, L 2 ); f Rn(L = ψ 1,L 2 ) n (f, L I 2 )( x n )d x n = n 1 I n 1 ( ) 1/2 f ( x n, x n ) 2 dx n d x n. I The Benedek-Panzone spaces R k (L 3, L ); f Rk (L 3,L ) = ( I n 1 [ ψk (f, L )( x k ) ]3 d x k ) 1/3 = ( I n 1 f ( x k, ) L (I ) 3 d xk ) 1/3.

6 Introduction 5/21 Mixed norm spaces Examples: The Lebesgue spaces L 1 (I) = R 1 (L 1, L 1 ); f R1(L 1,L 1 ) I = ψ 1 (f, L 1 )( x 1 )d x 1 = f ( x 1, x 1 ) dx 1 d x 1. n 1 I n 1 I The Benedek-Panzone spaces R n (L 1, L 2 ); f Rn(L = ψ 1,L 2 ) n (f, L I 2 )( x n )d x n = n 1 I n 1 ( ) 1/2 f ( x n, x n ) 2 dx n d x n. I The Benedek-Panzone spaces R k (L 3, L ); f Rk (L 3,L ) = ( I n 1 [ ψk (f, L )( x k ) ]3 d x k ) 1/3 = ( I n 1 f ( x k, ) L (I ) 3 d xk ) 1/3.

7 Introduction 5/21 Mixed norm spaces Examples: The Lebesgue spaces L 1 (I) = R 1 (L 1, L 1 ); f R1(L 1,L 1 ) I = ψ 1 (f, L 1 )( x 1 )d x 1 = f ( x 1, x 1 ) dx 1 d x 1. n 1 I n 1 I The Benedek-Panzone spaces R n (L 1, L 2 ); f Rn(L = ψ 1,L 2 ) n (f, L I 2 )( x n )d x n = n 1 I n 1 ( ) 1/2 f ( x n, x n ) 2 dx n d x n. I The Benedek-Panzone spaces R k (L 3, L ); f Rk (L 3,L ) = ( I n 1 [ ψk (f, L )( x k ) ]3 d x k ) 1/3 = ( I n 1 f ( x k, ) L (I ) 3 d xk ) 1/3.

8 Introduction 6/21 Mixed norm spaces Definition The mixed norm spaces R(X, Y ) are defined as follows n R(X, Y ) = R k (X, Y ). k=1 For each f R(X, Y ), we set f R(X,Y ) = n k=1 f Rk (X,Y ). Examples: The Lebesgue spaces L p (I n ) = R(L p, L p ), 1 p.

9 Introduction 7/21 Sobolev spaces We denote u = ( x1 u,..., xn u), where xi u is the distributional partial derivate of u with respect to x i. Definition The first-order Sobolev spaces are defined as W 1 Z(I n ) := { u L 1 loc (In ) : u Z(I n ) and u Z(I n ) }, with the norm u W 1 Z(I n ) = u Z(I n ) + u Z(I n ). By W 1 0 Z(In ) we denote the closure of C c (I n ) in W 1 Z(I n ).

10 Introduction 8/21 Classical Sobolev embedding theorem W 1 0 L p (I n ) L pn/(n p) (I n ), 1 p < n. Sobolev, case p > 1. His proof did not apply to p = 1. Gagliardo; Nirenberg, p = 1. W 1 0 L 1 (I n ) R(L 1, L ) L n (I n ). Fournier embedding theorem R(L 1, L ) L n,1 (I n ). W 1 0 L 1 (I n ) L n,1 (I n ) L n (I n ).

11 Sobolev embeddings in rearrangement-invariant Banach spaces 9/21 Sobolev embeddings in r.i. spaces Kerman and Pick studied the Sobolev embeddings among r.i. spaces. In particular, they solved the following problems: Given an r.i. range space X (I n ), find the largest r.i. domain space, with a.c. norm, namely Z(I n ), satisfying W 1 0 Z(I n ) X (I n ). This means that if W 1 0 Z(I n ) X (I n ) Z(I n ) Z(I n ). Given an r.i. domain space Z(I n ), describe the smallest r.i. range space, namely X (I n ), that verifies W 1 0 Z(I n ) X (I n ). That is, if W 1 0 Z(In ) X (I n ) X (I n ) X (I n ).

12 Sobolev embeddings in rearrangement-invariant Banach spaces 10/21 Examples Classical Sobolev embedding theorem W 1 0 L p (I n ) L pn/(n p) (I n ), 1 p < n. Hunt; O Neil; Peetre. W 1 0 L p (I n ) L pn/(n p),p (I n ). Kerman and Pick Optimal r.i. domain space: L p (I n ). Kerman and Pick Optimal r.i. range space: L pn/(n p),p (I n ).

13 Sobolev embeddings in rearrangement-invariant Banach spaces 11/21 Examples Critical Sobolev embedding theorem W 1 0 L n (I n ) L p (I n ), 1 p <. Maz ya; Hansson; Brézis and Wainger W 1 0 L n (I n ) L,n; 1 (I n ). Kerman and Pick Optimal r.i. domain space: Z L,n; 1(I n ). Hansson; Kerman and Pick Optimal r.i. range space: L,n; 1 (I n ).

14 Sobolev embeddings in mixed norm spaces 12/21 Motivation Classical Sobolev embeddings Gagliardo; Nirenberg W 1 0 L 1 (I n ) R(L 1, L ). Kerman and Pick Optimal domain and range of W0 1 Z(I n ) Y (I n ), within the class of r.i. spaces. Describe the largest domain space and the smallest range with mixed norm in W 1 0 Z(I n ) R(X, L ).

15 Sobolev embeddings in mixed norm spaces 13/21 Problem Let X (I n 1 ) be an r.i. space and let Z(I n ) be an r.i space, with a.c. norm. Our aim is to study the Sobolev embedding We are interested in the following questions: W 1 0 Z(I n ) R(X, L ). (1) Given a mixed norm space R(X, L ), we want to find the largest r.i. domain space, with a.c. norm, satisfying (1). Let Z(I n ) be an r.i. domain space, with a.c. norm. We would like to find the smallest range space of the form R(X, L ) for which (1) holds.

16 Sobolev embeddings in mixed norm spaces 14/21 Examples Classical Sobolev embedding theorem W 1 0 L p (I n ) L pn/(n p) (I n ), 1 p < n. W 1 0 L p (I n ) R(L p(n 1)/(n p),p, L ). L p (I n ) optimal r.i. domain space. R(L p(n 1)/(n p),p, L ) optimal range of the form R(X, L ).

17 Sobolev embeddings in mixed norm spaces 15/21 Examples Critical Sobolev embedding theorem W 1 0 L n (I n ) L p (I n ), 1 p <. W 1 0 L n (I n ) R(L,n; 1, L ). Z L,n; 1(I n ) optimal r.i. domain space. R(L,n; 1, L ) optimal range of the form R(X, L ).

18 Sobolev embeddings in mixed norm spaces 16/21 Domain space: L p (I n ), 1 p < n Classical Sobolev embedding theorem W 1 0 L p (I n ) L pn/(n p) (I n ), 1 p < n. Kerman and Pick L pn/(n p),p (I n ) optimal r.i. range space. R(L p(n 1)/(n p),p, L ) optimal range of the form R(X, L ). W 1 0 L p (I n ) R(L p(n 1)/(n p),p, L ) = L pn/(n p) (I n ).

19 Sobolev embeddings in mixed norm spaces 17/21 Domain space: L n (I n ) Critical Sobolev embedding theorem W 1 0 L n (I n ) L p (I n ), 1 p <. Kerman and Pick L,n; 1 (I n ) optimal r.i. range space. R(L,n; 1, L ) optimal range of the form R(X, L ). W 1 0 L n (I n ) R(L,n; 1, L ) = L,n; 1 (I n ).

20 Sobolev embeddings in mixed norm spaces 18/21 Domain space: Z(I n ) Z(I n ) r.i. domain space. X op (I n ) optimal r.i. range space for W 1 0 Z(I n ) X op (I n ). R(Y op, L ) optimal range of the form R(X, L ) for W 1 0 Z(I n ) R(Y op, L ). W 1 0 Z(I n ) R(Y op, L ) X op (I n ).

21 Critical case of the classical Sobolev embedding 19/21 Critical case of the classical Sobolev embedding Critical Sobolev embedding theorem Hansson; Kerman and Pick Optimal r.i range space W 1 0 L n (I n ) L,n; 1 (I n ). Bastero, Milman and Ruiz; Malý and Pick Improvement among non-linear r.i. spaces W 1 0 L n (I n ) L(, n)(i n ) = L,n; 1 (I n ).

22 Critical case of the classical Sobolev embedding 20/21 Critical case of the classical Sobolev embedding Critical Sobolev embedding theorem Optimal range of the form R(X, L ) W 1 0 L n (I n ) L,n; 1 (I n ). Improvement among non-linear spaces of the form R(X, L ) W 1 0 L n (I n ) R(L(, n), L ) = R(L,n; 1, L ).

23 The end Thank You!!

Poincaré inequalities that fail

ي ۆ Poincaré inequalities that fail to constitute an open-ended condition Lukáš Malý Workshop on Geometric Measure Theory July 14, 2017 Poincaré inequalities Setting Let (X, d, µ) be a complete metric