Three Extremal Problems in the Hardy and Bergman Spaces of Functions Analytic in a Disk

Let a nonnegativemeasurable function γ(ρ) be nonzero almost everywhere on (0, 1), and let the product ργ(ρ) be summable on (0, 1). Denote by B = Bγp, q, 1 ≤ p≤ ∞, 1 ≤ q < ∞, the space of functions f analytic in the unit disk for which the function Mpq (f, ρ)ργ(ρ) is summable on (0, 1), where Mp(f, ρ) is the p-mean of f on the circle of radius ρ; this space is equipped with the norm ||f||Bγp,q=||MP(f,.)||Lργ(p)q(0,1).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$||f||_{B_\gamma ^{p,q}} = ||{M_P}(f,.)||_{L_{\rho \gamma (p)}^q(0,1)}.$$\end{document} In the case q = ∞, the space B = Bγp, q is identified with the Hardy space Hp. Using an operator L given by the equality Lf(z)=∑k=0∞lkckzk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Lf(z) = \sum\nolimits_{k = 0}^\infty {{l_k}{c_k}{z^k}} $$\end{document} on functions f(z)=∑k=0∞ckzk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z) = \sum\nolimits_{k = 0}^\infty {{c_k}{z^k}} $$\end{document} analytic in the unit disk, we define the class LBγp,q(N):={f:||Lf||Bγp,q≤N},N>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$LB_\gamma^{p,q}(N) := \{f:||Lf||_{B_{\gamma}^{p,q}}\leq N \}, N > 0.$$\end{document}