Radial Two Weight Inequality for Maximal Bergman Projection Induced by a Regular Weight

Abstract

It is shown in quantitative terms that the maximal Bergman projection Pω+(f)(z)=∫Df(ζ)|Bzω(ζ)|ω(ζ)dA(ζ),\\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}$$ {P}^{+}_{\\omega}(f)(z)={\\int}_{\\mathbb{D}} f(\\zeta)|{B}^{\\omega}_{z}(\\zeta)|\\omega(\\zeta) dA(\\zeta), $$\\end{document} is bounded from L^{p}_{nu } to L^{p}_{eta } if and only if sup0<r<1∫0rη(s)∫s1ω(t)dtpds+11p∫r1ω(s)ν(s)1pp′ds1p′<∞,\\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}$$ \\underset{0<r<1}{\\sup}\\left( {\\int}_{0}^{r}\\frac{\\eta(s)}{\\left( {\\int}_{s}^{1}\\omega(t) dt\\right)^{p}} ds+1\\right)^{\\frac{1}{p}} \\left( {\\int}_{r}^{1}\\left( \\frac{\\omega(s)}{\\nu(s)^{\\frac{1}{p}}}\\right)^{p^{\\prime}}ds\\right)^{\\frac{1}{p^{\\prime}}}<\\infty, $$\\end{document} provided ω,ν,η are radial regular weights. A radial weight σ is regular if it satisfies {sigma }(r){asymp {int limits }}_{r}^{1}{sigma }(t) dt/(1-r) for all 0 ≤ r < 1. It is also shown that under an appropriate additional hypothesis involving ω and η, the Bergman projection Pω and {P}^{+}_{omega } are simultaneously bounded.