Haar system as Schauder basis in Besov spaces: The limiting cases for 0 < p <= 1
Settles a long-standing open problem from 1979 for the one-dimensional case and provides complete characterization for higher dimensions, benefiting researchers in functional analysis and approximation theory.
This paper resolves the open case of whether the Haar system is a Schauder basis in Besov spaces B_{p,q,1}^s for 0<p<1 in the limiting case s=d(1/p-1), proving it holds if and only if 0<q≤p. It also extends results to standard Besov spaces B_{p,q}^s for limiting cases s=d(1/p-1) and s=1.
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].