Bergman Projection on Lebesgue Space Induced by Doubling Weight

Abstract

Let ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} and ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} be radial weights on the unit disc of the complex plane, and denote σ=ωp′ν-p′p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma =\\omega ^{p'} \ u ^{-\\frac{p'}{p}}$$\\end{document} and ωx=∫01sxω(s)ds\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _x =\\int _0^1\\,s^x \\omega (s)\\,ds$$\\end{document} for all 1≤x<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le x<\\infty $$\\end{document}. Consider the one-weight inequality ‖Pω(f)‖Lνp≤C‖f‖Lνp,1<p<∞,(†)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert P_\\omega (f)\\Vert _{L^p_\ u }\\le C\\Vert f\\Vert _{L^p_\ u },\\quad 1<p<\\infty , \\qquad \\qquad (\\dagger ) \\end{aligned}$$\\end{document}for the Bergman projection Pω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_\\omega $$\\end{document} induced by ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}. It is shown that the moment condition Dp(ω,ν)=supn∈N∪{0}νnp+11pσnp′+11p′ω2n+1<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} D_p(\\omega ,\ u )=\\sup _{n\\in {\\mathbb {N}}\\cup \\{0\\}} \\frac{\\left( \ u _{np+1}\\right) ^\\frac{1}{p}\\left( \\sigma _{np'+1} \\right) ^\\frac{1}{p'}}{\\omega _{2n+1}}<\\infty \\end{aligned}$$\\end{document}is necessary for (†\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dagger $$\\end{document}) to hold. Further, Dp(ω,ν)<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_p(\\omega ,\ u )<\\infty $$\\end{document} is also sufficient for (†\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dagger $$\\end{document}) if ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} admits the doubling properties sup0≤r<1∫r1ν(s)sds∫1+r21ν(s)sds<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sup _{0\\le r<1}\\frac{\\int _r^1\ u (s) s\\,ds}{\\int _{\\frac{1+r}{2}}^1\ u (s)s\\,ds}<\\infty $$\\end{document} and sup0≤r<1∫r1ν(s)sds∫r1-1-rKν(s)sds<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sup _{0\\le r<1}\\frac{\\int _r^1\ u (s)s\\,ds}{\\int _r^{1-\\frac{1-r}{K}} \ u (s)s\\,ds}<\\infty $$\\end{document} for some K>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K>1$$\\end{document}. In addition, an analogous result for the one weight inequality ‖Pω(f)‖Dν,kp≤C‖f‖Lνp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert P_\\omega (f)\\Vert _{D^p_{\ u ,k}} \\le C\\Vert f\\Vert _{L^p_\ u }$$\\end{document}, where ‖f‖Dν,kpp=∑j=0k-1|f(j)(0)|p+∫D|f(k)(z)|p(1-|z|)kpν(z)dA(z)<∞,k∈N,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert f \\Vert _{D^p_{\ u , k}}^p =\\sum \\limits _{j=0}^{k-1}\\vert f^{(j)}(0)\\vert ^p +\\int _{{\\mathbb {D}}} \\vert f^{(k)}(z)\\vert ^p (1-\\vert z \\vert )^{kp} \ u (z)\\,dA(z)<\\infty , \\quad k\\in {\\mathbb {N}}, \\end{aligned}$$\\end{document}is established. The inequality (†\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dagger $$\\end{document}) is further studied by using the necessary condition Dp(ω,ν)<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_p(\\omega ,\ u )<\\infty $$\\end{document} in the case of the exponential type weights ν(r)=exp-α(1-rl)β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u (r)=\\exp \\left( -\\frac{\\alpha }{(1-r^l)^{\\beta }} \\right) $$\\end{document} and ω(r)=exp-α~(1-rl~)β~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega (r)= \\exp \\left( -\\frac{ \\widetilde{\\alpha }}{(1-r^{\\widetilde{l}})^{\\widetilde{\\beta }}} \\right) $$\\end{document}, where 0<α,α~,l,l~<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\alpha , \\, \\widetilde{\\alpha }, \\, l, \\, \\widetilde{l}<\\infty $$\\end{document} and 0<β,β~≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\beta , \\, \\widetilde{\\beta }\\le 1$$\\end{document}.