Basis properties of the Haar system in various function spaces, I. Andreas Seeger (University of Wisconsin-Madison) Chemnitz Summer School on Applied Analysis 2019 • Based on joint work with Gustavo Garrigós and Tino Ullrich
• A.S., T. Ullrich. Haar projection numbers and failure of unconditional convergence in Sobolev spaces. Mathematische Zeitschrift, 285 (2017), 91-119. • ..., Lower bounds for Haar projections: Deterministic Examples. Constructive Approximation, 42 (2017), 227-242. • G. Garrigós, A.S. and T. Ullrich. The Haar system as a Schauder basis in spaces of Hardy-Sobolev type. Journal of Fourier Analysis and Applications, 24(5) (2018), 1319-1339. • ..., Basis properties of the Haar system in limiting Besov spaces. Preprint (arXiv). • ..., The Haar system in Triebel-Lizorkin spaces: Endpoint cases. Preprint (arXiv).
The Haar system H Haar (1910): For j ∈ N 0 , µ ∈ Z let h j ,µ be supported on I j ,µ = [ 2 − j µ, 2 − j ( µ + 1 )) and � 1 on the left half of I j ,µ h j ,µ ( x ) = − 1 on the right half of I j ,µ • The Haar frequency of h j ,µ is 2 j . • The functions 2 j / 2 h j ,µ , together with the functions h − 1 ,µ := 1 [ µ,µ + 1 ) form an ONB of L 2 ( R ) . • Let H be the collection of h j ,µ , j = − 1 , 0 , 1 , 2 , . . . , µ ∈ Z . Haar system on [ 0 , 1 ) , or T : Take only those Haar functions defined on [ 0 , 1 ) .
Haar system in d dimensions • Intervals are replaced by cubes. For every dyadic cube we have 2 d − 1 Haar functions. u ( 0 ) = ✶ [ 0 , 1 ) , u ( 1 ) = ✶ [ 0 , 1 / 2 ) − ✶ [ 1 / 2 , 1 ) . Let For every ε = ( ε 1 , . . . , ε d ) ∈ { 0 , 1 } d let h ( ε ) ( x 1 , . . . , x d ) = u ( ε 1 ) ( x 1 ) · · · u ( ε d ) ( x d ) . Finally, one sets h ( ε ) j ,ℓ ( x ) = h ( ε ) ( 2 j x − ℓ ) , j ∈ Z , ℓ ∈ Z d , The Haar system H d is then given by � � � � h ( � j ,ℓ | j ∈ Z , ℓ ∈ Z d , ε ∈ { 0 , 1 } d \ { � 0 ) h ( ε ) H d = ℓ ∈ Z d ∪ 0 } . 0 ,ℓ
Bases, I Def. 1 Given a (quasi-)Banach space X of tempered distributions in R d and an enumeration U = ( u 1 , u 2 , . . . ) of the Haar system H d , we say that U is a basic sequence on X if the orthogonal projections P n : span ( U ) → span ( { u 1 , . . . , u n } ) are uniformly bounded. • Only seemingly weaker: Any f in the closure of span ( U ) can be expanded in a unique way as � f = c n ( f ) u n n with convergence in X . Then c n ( f ) = 2 freq ( u n ) � f , u n � u n . Def. 2. If U is a basic sequence on X and if span ( U ) is dense in X then we say that U is a Schauder basis of X .
Bases, II Assume that span ( U ) is dense in X and suppose that U is a Schauder basis Def. 3. U is an unconditional basis of X if for f = � n c n u n we have that ∞ � c ̟ ( n ) u ̟ ( n ) n = 1 converges for every bijection ̟ : N → N . Equivalently: � ∞ n = 1 ± c n u n converges for all choices of ± 1. � ∞ n = 1 ± m ( n ) c n u n converges for all m ∈ ℓ ∞ ( N ) .
Bases, III Use the UBP: • U is an unconditional basis if and only if the span of U is dense and if the projections to subspaces generated by finite subsets of U are uniformly bounded. • For unconditional bases the multiplier problem is trivial: U is an unconditional basis if and only if the multiplier transformation � � f = c n ( f ) u n �→ m ( n ) c n ( f ) u n n n is a bounded operator for all m ∈ ℓ ∞ ( N ) .
Bases, IV We say that U is an unconditional basic sequence on X if U is X . an unconditional basis on span ( U ) For the Haar system the following notion is also useful. Def. H d is a local basis of X if f = � c n ( f ) u n converges for all compactly supported f . The statements about projection operators remain true, but the operator norms depend on the choice of a compact set K in which all considered f are supported. One can also define the notions of local unconditional basis, local basic sequence and local unconditional basic sequence.
The Haar basis in L p ( R ) Schauder (28): H (with the natural lexicographic order) is a basis of L p ([ 0 , 1 )) when 1 ≤ p < ∞ . 2 j − 1 ∞ � � 2 j � f , h j ,µ � h j ,µ f = E 0 f + j = 0 µ = 0 for f ∈ L p ([ 0 , 1 )) , with convergence in L p . • One works with conditional expectional operators E N associated to dyadic intervals of length 2 − N . • E N + 1 − E N is the orthogonal projection to the space generated by the Haar functions with Haar frequency 2 N . • Billard (1970’s): H is a Schauder basis on the Hardy space h 1 ( T ) .
The Haar basis in L p ( R ) , 1 < p < ∞ Marcinkiewicz (37): H is an unconditional basis of L p ( R ) when 1 < p < ∞ . • For f ∈ L p , ∞ � � 2 j � f , h j ,µ � h j ,µ f = j = − 1 µ ∈ Z with unconditional convergence in L p . Based on prior work of Paley, on square functions. • Pełczynski (61): L 1 cannot be imbedded in a Banach space with an unconditional basis.
Function spaces on R d , I. Sobolev spaces W m p , 1 ≤ p ≤ ∞ , m ∈ N . � � ∂ α f � p � f � W m p = | α |≤ m Bessel potential space L p s aka Sobolev space. s = � ( I − ∆) s / 2 f � p � f � L p where F [( I − ∆) s / 2 f ]( ξ ) = ( 1 + | ξ | 2 ) s / 2 � f ( ξ ) . Note that for 1 < p < ∞ we have W p s = L p s and the L p s interpolate with the complex method. Since Haar functions are not smooth we are interested in these spaces for small s .
Function spaces on R d , II. L p Hölder classes Λ( p , s ) ≡ B s p , ∞ For 0 < s < 1, 1 ≤ p ≤ ∞ , � f ( · + h ) − f � p � f � B s p , ∞ = � f � p + sup . | h | s h � = 0 Sobolev-Slobodecki spaces B s p , p . For 0 < s < 1 let � �� | f ( x ) − f ( y ) | p � 1 / p � f � B s p , p = � f � p + | x − y | d + sp dx dy B s p , p is also referred to as "Sobolev space of fractional order s ". p , p � = L p But B s s for p � = 2.
Function spaces, III. The role of square functions Consider { P k } ∞ k = 0 , an inhomogeneous dyadic frequency decomposition. Aka Littlewood-Paley decomposition. Let φ 0 ∈ C ∞ c (( R d ) ∗ ) , φ 0 = 1 near 0. P 0 f ( ξ ) = φ 0 ( ξ ) � � f ( ξ ) , P k f ( ξ ) = ( φ 0 ( 2 − k ξ ) − φ 0 ( 2 1 − k ξ )) � � f ( ξ ) , k ≥ 1 . • Localization to frequencies of size ≈ 2 k . Then, for 1 < p < ∞ � 2 2 ks | P k f | 2 � 1 / 2 � � ∞ � � � � f � L p s = � � p . k = 0 by standard singular integral theory (in a Hilbert-space setting).
Function spaces, IV. B s p , q , F s p , q "Function spaces" as subspaces of tempered distributions via Fourier analytic definitions: Besov-Nikolskij-Taibleson spaces, 0 < p , q ≤ ∞ . � � � { 2 ks P k f } � � f � B s p , q = ℓ q ( L p ) Triebel-Lizorkin spaces. 0 < p < ∞ , 0 < q ≤ ∞ . � � � { 2 ks P k f } � � f � F s p , q = L p ( ℓ q ) p , 2 = L p Note F s s , 1 < p < ∞ . Hardy-Sobolev H s p when p > 0. There is an extension to p = ∞ , so that F 0 ∞ , 2 = BMO , using BMO -like norms in the general case (cf. the Chang-Wilson-Wolff theorem and Frazier-Jawerth definitions).
Function spaces, V. Peetre maximal functions Motivated by the Hardy space results of Fefferman-Stein, Peetre (1975) introduced maximal functions on distributions with bounded Fourier theorems about maximal functions support. Assume f Schwartz and � f supported in a set of diameter 1. Then � � 1 / r . | f ( x + h ) | M HL [ | f | r ] sup ( 1 + | h | ) d / r � x ∈ R d Summary of proof: One proves first |∇ f ( x + h ) | | f ( x + h ) | sup ( 1 + | h | ) d / r � sup ( 1 + | h | ) d / r x ∈ R d x ∈ R d and then relies on a mean value inequality � av B δ ( x ) | g | r � 1 / r | g ( x ) | ≤ c 1 δ sup |∇ g | + c 2 . B δ ( x )
Function spaces, VI. Peetre maximal functions: Scaled and vector valued versions • Assume that � f k ∈ S ′ is supported on a set of diameter R k . Let | f k ( x + h ) | M k , A f k ( x ) = sup ( 1 + R k | h | ) A . h ∈ R d Then � � � � � {M k , A f k } � � { f k } � ℓ q ( L p ) , A > d / p . ℓ q ( L p ) � A � � � � � {M k , A f k } � � { f k } � L p ( ℓ q ) � A L p ( ℓ q ) , A > d / p , A > d / q . One can use Fefferman-Stein vector-valued extension of the Hardy-Littlewood maximal theorem. Is the additional condition on q needed?
Function spaces, VI. Peetre maximal functions: Scaled and vector valued versions • Assume that � f k ∈ S ′ is supported on a set of diameter R k . Let | f k ( x + h ) | M k , A f k ( x ) = sup ( 1 + R k | h | ) A . h ∈ R d Then � � � � � {M k , A f k } � � { f k } � ℓ q ( L p ) , A > d / p . ℓ q ( L p ) � A � � � � � {M k , A f k } � � { f k } � L p ( ℓ q ) � A L p ( ℓ q ) , A > d / p , A > d / q . One can use Fefferman-Stein vector-valued extension of the Hardy-Littlewood maximal theorem. Is the additional condition on q needed? Yes, no matter what the R k are. (Christ, S., PLMS 2006).
Recommend
More recommend
