Title: Basis properties for the Haar system in Hardy-Sobolev spaces
Abstract: We give a complete description of the Schauder basis property for the classical Haar system in the Hardy-Sobolev spaces $H^s_p(\mathbb{R}^d)$ for all $s\in \mathbb{R}$ and $p\in (0,\infty)$. In particular, when $d=1$, the property holds in the open range of indices $$1/p -1 < s < \min \{1/p, 1\},$$ while unconditionality additionally requires $-1/2 < s < 1/2$. We also establish the behavior at the end-point cases, obtaining a full description in the setting of Besov and Triebel-Lizorkin spaces, $B^s_{p,q}$ and $F^s_{p,q}$, where the validity of the property depends on the index $q$.
The approach is based in finding suitable uniform bounds for the dyadic averaging operators in such spaces. As a consequence we obtain new sufficient conditions for certain Haar multipliers in the range of indices where unconditionality fails.
These results are part of a joint work with Andreas Seeger and Tino Ullrich.
Slides