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 $1<p<\infty$ 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.
Categories: math.CA math.FA