A Note on Mixed Norm Spaces Nadia Clavero University of Barcelona - PowerPoint PPT Presentation

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

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

Introduction Rearrangement-invariant Banach function spaces n � �� � Given a finite interval I , we denote I n = I × . . . × I , n ∈ N . Definition 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  � � � 1 / p ,  I n | f ( x ) | p dx 1 ≤ p < ∞ ; � � � f � L p ( I n ) = � �  inf C ≥ 0 : | f ( x ) | ≤ C a.e , p = ∞ . 3/21

Introduction 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 � � � � f ∈ M ( I n ) : � f � R k ( X , Y ) = R k ( X , Y ) < ∞ , � � � � � � � f � � ψ k ( f , Y ) � � f ( � � where R k ( X , Y ) = X ( I n − 1 ) , ψ k ( f , Y )( � x k ) = x k , · ) Y ( I ) . 4/21

Introduction Mixed norm spaces Examples: The Lebesgue spaces L 1 ( I ) = R 1 ( L 1 , L 1 ); � � � � � � f � I n − 1 ψ 1 ( f , L 1 )( � R 1 ( L 1 , L 1 ) = x 1 ) d � x 1 = | f ( � x 1 , x 1 ) | dx 1 d � x 1 . I n − 1 I The Benedek-Panzone spaces R n ( L 1 , L 2 ); � � � � � 1 / 2 � � � f � I n − 1 ψ n ( f , L 2 )( � x n , x n ) | 2 dx n R n ( L 1 , L 2 ) = x n ) d � x n = | f ( � d � x n . I n − 1 I The Benedek-Panzone spaces R k ( L 3 , L ∞ ); � � � 1 / 3 � � � 1 / 3 � � � � 3 d � � � 3 � f � ψ k ( f , L ∞ )( � � f ( � � R k ( L 3 , L ∞ ) = x k ) x k = x k , · ) d � x k . L ∞ ( I ) I n − 1 I n − 1 5/21

Introduction Mixed norm spaces Examples: The Lebesgue spaces L 1 ( I ) = R 1 ( L 1 , L 1 ); � � � � � � f � I n − 1 ψ 1 ( f , L 1 )( � R 1 ( L 1 , L 1 ) = x 1 ) d � x 1 = | f ( � x 1 , x 1 ) | dx 1 d � x 1 . I n − 1 I The Benedek-Panzone spaces R n ( L 1 , L 2 ); � � � � � 1 / 2 � � � f � I n − 1 ψ n ( f , L 2 )( � x n , x n ) | 2 dx n R n ( L 1 , L 2 ) = x n ) d � x n = | f ( � d � x n . I n − 1 I The Benedek-Panzone spaces R k ( L 3 , L ∞ ); � � � 1 / 3 � � � 1 / 3 � � � � 3 d � � � 3 � f � ψ k ( f , L ∞ )( � � f ( � � R k ( L 3 , L ∞ ) = x k ) x k = x k , · ) d � x k . L ∞ ( I ) I n − 1 I n − 1 5/21

Introduction Mixed norm spaces Examples: The Lebesgue spaces L 1 ( I ) = R 1 ( L 1 , L 1 ); � � � � � � f � I n − 1 ψ 1 ( f , L 1 )( � R 1 ( L 1 , L 1 ) = x 1 ) d � x 1 = | f ( � x 1 , x 1 ) | dx 1 d � x 1 . I n − 1 I The Benedek-Panzone spaces R n ( L 1 , L 2 ); � � � � � 1 / 2 � � � f � I n − 1 ψ n ( f , L 2 )( � x n , x n ) | 2 dx n R n ( L 1 , L 2 ) = x n ) d � x n = | f ( � d � x n . I n − 1 I The Benedek-Panzone spaces R k ( L 3 , L ∞ ); � � � 1 / 3 � � � 1 / 3 � � � � 3 d � � � 3 � f � ψ k ( f , L ∞ )( � � f ( � � R k ( L 3 , L ∞ ) = x k ) x k = x k , · ) d � x k . L ∞ ( I ) I n − 1 I n − 1 5/21

Introduction 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 � � � � R ( X , Y ) = � n � f � � f � For each f ∈ R ( X , Y ) , we set R k ( X , Y ) . k =1 Examples: The Lebesgue spaces L p ( I n ) = R ( L p , L p ) , 1 ≤ p ≤ ∞ . 6/21

Introduction Sobolev spaces We denote ∇ u = ( ∂ x 1 u , . . . , ∂ x n u ) , where ∂ x i 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 ( I n ) : 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 c ( I n ) in W 1 Z ( I n ) . 0 Z ( I n ) we denote the closure of C ∞ 7/21

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

Sobolev embeddings in rearrangement-invariant Banach spaces 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 ) . 0 � → X ( I n ) ⇒ � This means that if W 1 Z ( I n ) ֒ Z ( I n ) ֒ → Z ( I n ) . domain space Z ( I n ), describe the smallest r.i. ∗ Given an 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 ( I n ) ֒ X ( I n ) ⇒ X ( I n ) ֒ X ( I n ) . 9/21

Sobolev embeddings in rearrangement-invariant Banach spaces 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 Kerman and Pick Optimal r.i. domain space: L p ( I n ) . Optimal r.i. range space: L pn / ( n − p ) , p ( I n ) . 10/21

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

Sobolev embeddings in mixed norm spaces Motivation Classical Sobolev embeddings Kerman and Pick Gagliardo; Nirenberg W 1 0 L 1 ( I n ) ֒ → R ( L 1 , L ∞ ) . Optimal domain and range of W 1 0 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 ∞ ) . 12/21

Sobolev embeddings in mixed norm spaces 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 W 1 0 Z ( I n ) ֒ → R ( X , L ∞ ) . (1) We are interested in the following questions: 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. 13/21

Sobolev embeddings in mixed norm spaces 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 ∞ ) . R ( L p ( n − 1) / ( n − p ) , p , L ∞ ) optimal L p ( I n ) optimal r.i. domain space. range of the form R ( X , L ∞ ) . 14/21

Sobolev embeddings in mixed norm spaces 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 ∞ ) . R ( L ∞ , n ; − 1 , L ∞ ) optimal Z L ∞ , n ; − 1 ( I n ) optimal r.i. domain space. range of the form R ( X , L ∞ ) . 15/21

Sobolev embeddings in mixed norm spaces 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 . R ( L p ( n − 1) / ( n − p ) , p , L ∞ ) optimal Kerman and Pick L pn / ( n − p ) , p ( I n ) optimal r.i. range space. range of the form R ( X , L ∞ ) . W 1 → R ( L p ( n − 1) / ( n − p ) , p , L ∞ ) ֒ � = L pn / ( n − p ) ( I n ) . 0 L p ( I n ) ֒ → 16/21

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

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