Multivariate Haar systems in Besov function spaces

Abstract

We determine all cases for which the $d$-dimensional Haar wavelet system $H^d$ on the unit cube $I^d$ is a conditional or unconditional Schauder basis in the classical isotropic Besov function spaces ${B}_{p,q,1}^s(I^d)$, $0<p,q<\infty$, $0\le s < 1/p$, defined in terms of first-order $L_p$ moduli of smoothness. We obtain similar results for the tensor-product Haar system $\tilde{H}^d$, and characterize the parameter range for which the dual of ${B}_{p,q,1}^s(I^d)$ is trivial for $0<p<1$.