Abstract
We show that, for suitable enumerations, the Haar system is a Schauder basis in the classical Sobolev spaces in \({\mathbb R}^d\) with integrability \(1<p<\infty \) and smoothness \(1/p-1<s<1/p\). This complements earlier work by the last two authors on the unconditionality of the Haar system and implies that it is a conditional Schauder basis for a nonempty open subset of the (1 / p, s)-diagram. The results extend to (quasi-)Banach spaces of Hardy–Sobolev and Triebel–Lizorkin type in the range of parameters \(\frac{d}{d+1}<p<\infty \) and \(\max \{d(1/p-1),1/p-1\}<s<\min \{1,1/p\}\), which is optimal except perhaps at the end-points.
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Acknowledgements
The authors worked on this paper while participating in the 2016 summer program in Constructive Approximation and Harmonic Analysis at the Centre de Recerca Matemàtica at the Universitat Autònoma de Barcelona, Spain. They would like to thank the organizers of the program for providing a pleasant and fruitful research atmosphere. We also thank the referee for various useful comments that have led to an improved version of this paper. Finally, T.U. thanks Peter Oswald for discussions concerning [8] and the results in Sect. 4. G.G. was supported in part by Grants MTM2013-40945-P, MTM2014-57838-C2-1-P, MTM2016-76566-P from MINECO (Spain), and Grant 19368/PI/14 from Fundación Séneca (Región de Murcia, Spain). A.S. was supported in part by NSF Grant DMS 1500162. T.U. was supported the DFG Emmy-Noether program UL403/1-1.
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Communicated by Vladimir Temlyakov.
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Garrigós, G., Seeger, A. & Ullrich, T. The Haar System as a Schauder Basis in Spaces of Hardy–Sobolev Type. J Fourier Anal Appl 24, 1319–1339 (2018). https://doi.org/10.1007/s00041-017-9583-1
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DOI: https://doi.org/10.1007/s00041-017-9583-1