Abstract
Let {mathbb {D}} be the open unit disk of the complex plane {mathbb {C}} and H({mathbb {D}} ) be the space of all analytic functions on {mathbb {D}} . Let A^{2}_{gamma ,delta }({mathbb {D}} ) be the space of analytic functions that are L^{2} with respect to the weight omega _{gamma ,delta }(z)=( ln frac{1}{|z|})^{gamma }[ln (1-frac{1}{ln |z|})]^{delta }, where -1<gamma <infty and delta le 0. For given gin H({mathbb {D}} ), the integral-type operator I_{g} on H({mathbb {D}} ) is defined as \t\t\tIgf(z)=∫0zf(ζ)g(ζ)dζ.\\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}$$ I_{g}f(z)= \\int _{0}^{z}f(\\zeta )g(\\zeta )\\,d\\zeta . $$\\end{document} In this paper, we characterize the boundedness of I_{g} on A^{2}_{gamma ,delta }, whereas in the main result we estimate the essential norm of the operator. Some basic results on the space A^{2}_{gamma ,delta }({mathbb {D}} ) are also presented.
Highlights
Let D = {z ∈ C : |z| < 1} be the open unit disk of the complex plane C, rD = {z ∈ D : |z| < r}, H(D)be the space of all analytic functions on D, and dA(z) =1 π r dr dθ normalized area measure on D (i.e., A(D) = 1).A positive continuous function on D is called weight
The weighted-type space Hμ∞(D) = Hμ∞ consists of f ∈ H(D) such that f Hμ∞ := sup μ(z) f (z) < ∞
The little weighted-type space on D, Hμ∞,0(D) = Hμ∞,0 consists of all f ∈ H(D) such that lim μ(z) f (z) = 0
Summary
Let D = {z ∈ C : |z| < 1} be the open unit disk of the complex plane C, rD = {z ∈ D : |z| < r}, H(D). 1 π r dr dθ normalized area measure on D (i.e., A(D) = 1). A positive continuous function on D is called weight. Let μ(z) be a weight function on D. The weighted-type space Hμ∞(D) = Hμ∞ consists of f ∈ H(D) such that f Hμ∞ := sup μ(z) f (z) < ∞. The little weighted-type space on D, Hμ∞,0(D) = Hμ∞,0 consists of all f ∈ H(D) such that lim μ(z) f (z) = 0. For μ(z) = (1 – |z|)α, α > 0, the classical weighted-type space Hα∞(D) = Hα∞ and the classical little weighted-type space Hα∞,0(D) =
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