The Haar system as a Schauder basis in spaces of Hardy-Sobolev type

We show that, for suitable enumerations, the multivariate Haar system is a Schauder basis in the classical Sobolev spaces on ${\mathbb{R}}^d$ with integrability $1<p<\mathrm{\infty }$ and smoothness $1\mathrm{/}p-1<s<1\mathrm{/}p$. 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 $(1\mathrm{/}p,s)$-diagram. The results extend to (quasi-)Banach spaces of Hardy-Sobolev and Triebel-Lizorkin type in the range of parameters $\frac{d}{d+1}<p<\mathrm{\infty }$ and $\max \{d(1\mathrm{/}p-1),1\mathrm{/}p-1\}<s<\min \{1,1\mathrm{/}p\}$, which is optimal except perhaps at the end-points.