Besov Spaces Induced by Doubling Weights

Abstract

Let 1leqslant p<infty , 0<q<infty , and nu be a two-sided doubling weight satisfying sup0⩽r<1(1-r)q∫r1ν(t)dt∫0rν(s)(1-s)qds<∞.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sup _{0\\leqslant r<1}\\frac{(1-r)^q}{\\int _r^1\\nu (t)\\,dt}\\int _0^r\\frac{\\nu (s)}{(1-s)^q}\\,ds<\\infty . \\end{aligned}$$\\end{document}The weighted Besov space mathcal {B}_{nu }^{p,q} consists of those fin H^p such that ∫01∫02π|f′(reiθ)|pdθq/pν(r)dr<∞.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\int _0^1 \\left( \\int _{0}^{2\\pi } |f'(re^{i\\theta })|^p\\,d\\theta \\right) ^{q/p}\\nu (r)\\,dr<\\infty . \\end{aligned}$$\\end{document}Our main result gives a characterization for fin mathcal {B}_{nu }^{p,q} depending only on |f|, p, q, and nu . As a consequence of the main result and inner-outer factorization, we obtain several interesting by-products. For instance, we show the following modification of a classical factorization by F. and R. Nevanlinna: If fin mathcal {B}_{nu }^{p,q}, then there exist f_1,f_2in mathcal {B}_{nu }^{p,q} cap H^infty such that f=f_1/f_2. Moreover, we give a sufficient and necessary condition guaranteeing that the product of fin H^p and an inner function belongs to mathcal {B}_{nu }^{p,q}. Applying this result, we make some observations on zero sets of mathcal {B}_{nu }^{p,p}.