The Haar System as a Schauder Basis in Spaces of Hardy–Sobolev Type

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.