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arXiv · 1808.08156

Haar system as Schauder basis in Besov spaces: The limiting cases for 0 < p <= 1

Peter Oswald
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Abstract

We show that the d-dimensional Haar system H^d on the unit cube I^d is a Schauder basis in the classical Besov space B_{p,q,1}^s(I^d), 0<p<1, defined by first order differences in the limiting case s=d(1/p-1), if and only if 0<q\le p. For d=1 and p<q, this settles the only open case in our 1979 paper [4], where the Schauder basis property of H in B_{p,q,1}^s(I) for 0<p<1 was left undecided. We also consider the Schauder basis property of H^d for the standard Besov spaces B_{p,q}^s(I^d) defined by Fourier-analytic methods in the limiting cases s=d(1/p-1) and s=1, complementing results by Triebel [7].

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Peter Oswald. 2018-08-24. Haar system as Schauder basis in Besov spaces: The limiting cases for 0 < p <= 1. https://arxiv.org/abs/1808.08156

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