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

Abstract

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