Haar projection numbers and failure of unconditional convergence in Sobolev spaces

For $$1<p<\infty $$1<p<∞ we determine the precise range of $$L_p$$Lp Sobolev spaces for which the Haar system is an unconditional basis. We also consider the natural extensions to Triebel–Lizorkin spaces and prove upper and lower bounds for norms of projection operators depending on properties of the Haar frequency set.

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