Haar frame characterizations of Besov–Sobolev spaces and optimal embeddings into their dyadic counterparts

Garrigós Aniorte, Gustavo Adolfo; Seeger, Andreas; Ullrich, Tino

Publication:
Haar frame characterizations of Besov–Sobolev spaces and optimal embeddings into their dyadic counterparts

dc.contributor.author	Garrigós Aniorte, Gustavo Adolfo
dc.contributor.author	Seeger, Andreas
dc.contributor.author	Ullrich, Tino
dc.contributor.department	Matemáticas
dc.date.accessioned	2026-01-25T09:10:34Z
dc.date.available	2026-01-25T09:10:34Z
dc.date.copyright	© 2023, The Author(s)
dc.date.issued	2023
dc.description.abstract	We study the behavior of Haar coefficients in Besov and Triebel-Lizorkin spaces on R , for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness s<1, in which the spaces F(s,p,q) and B(s,p,q) are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that 1/p<s<1 and f in B(s,p,q), we actually prove that the usual Haar coefficient norm of { 2^j(f,h_{j,m}) } in b(s,p,q) remains equivalent to the norm of f in B(s,p,q), i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case s=1 and q=infty, we show that such an expression gives an equivalent norm for the Sobolev space W(1,p) in R , for 1<p<infty, which is related to a classical result by Bočkarev. Finally, in several endpoint cases we clarify the relation between dyadic and standard Besov and Triebel-Lizorkin spaces.
dc.format	application/pdf
dc.format.extent	45
dc.identifier.citation	J Fourier Anal Appl 29, 39 (2023)
dc.identifier.doi	https://doi.org/10.1007/s00041-023-10013-7
dc.identifier.eissn	1531-5851
dc.identifier.issn	1069-5869
dc.identifier.uri	http://hdl.handle.net/10201/193169
dc.language	eng
dc.publisher	Springer
dc.relation	G.G. was supported in part by grant PID2019-105599GB- I00 from Ministerio de Ciencia e Innovación (Spain), and grant 20906/PI/18 from Fundación Séneca (Región de Murcia, Spain). A.S. was supported in part by National Science Foundation grant DMS 2054220. T.U. was supported in part by Deutsche Forschungsgemeinschaft (DFG), grant 403/2-1. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
dc.relation.publisherversion	https://link.springer.com/article/10.1007/s00041-023-10013-7
dc.relation.replaces	https://arxiv.org/abs/2209.02630
dc.rights	Attribution-NonCommercial-NoDerivatives 4.0 International	*
dc.rights.accessRights	info:eu-repo/semantics/openAccess
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/4.0/	*
dc.subject	Haar system
dc.subject	Haar frames
dc.subject	Sobolev spaces
dc.subject	Besov and Triebel Lizorkin spaces
dc.subject	Dyadic versions of function spaces
dc.subject	Wavelets
dc.subject	Splines
dc.subject.ods	No relacionado con ningún objetivo de desarrollo sostenible
dc.title	Haar frame characterizations of Besov–Sobolev spaces and optimal embeddings into their dyadic counterparts
dc.type	info:eu-repo/semantics/article
dc.type.version	info:eu-repo/semantics/acceptedVersion
dspace.entity.type	Publication	es
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