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

Abstract

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 = B_γ^p,
q, 1 ≤ p≤ ∞, 1 ≤ q < ∞, the space of functions f analytic in the unit disk for which the function M_p^q (f, ρ)ργ(ρ) is summable on (0, 1), where M_p(f, ρ) is the p-mean of f on the circle of radius ρ; this space is equipped with the norm

\(||f||_{B_\gamma ^{p,q}} = ||{M_P}(f,.)||_{L_{\rho \gamma (p)}^q(0,1)}.\)

In the case q = ∞, the space B = B_γ^p,
q is identified with the Hardy space H^p. Using an operator L given by the equality \(Lf(z) = \sum\nolimits_{k = 0}^\infty {{l_k}{c_k}{z^k}} \) on functions \(f(z) = \sum\nolimits_{k = 0}^\infty {{c_k}{z^k}} \) analytic in the unit disk, we define the class

\(LB_\gamma^{p,q}(N) := \{f:||Lf||_{B_{\gamma}^{p,q}}\leq N \}, N > 0.\)

For a pair of such operators L and G, under some constraints, the following three extremal problems are solved.

1. The best approximation of the class (1) by the class \(LB_\gamma ^{{p_1},{q_1}}(1)\) by the class \(GB_\gamma^{p_{3}, q_{3}}(N)\) in the norm of the space \(B_\gamma^{p_{2}, q_{2}}\) is found for 2 ≤ p₁ ≤ ∞, 1 ≤ p₂ ≤ 2, 1 ≤ p₃ ≤ 2, 1 ≤ q₁ = q₂ = q₃ ≤ ∞, and q_s = 2 or ∞.

2. The best approximation of the operator L by the set L(N), N > 0, of bounded linear operators from \(B_\gamma ^{{p_1},{q_1}}\) to \(B_\gamma ^{{p_2},{q_2}}\) with the norm not exceeding N on the class \(GB_\gamma ^{{p_3},{q_3}}\) (1) is found for 2 ≤ p₁ ≤∞, 1 ≤ p₂ ≤ 2, 2 ≤ p₃ ≤ ∞, 1 ≤ q₁ = q₂ = q₃ ≤ ∞, and q_s = 2 or ∞.

3. Bounds for the modulus of continuity of the operator L on the class \(GB_\gamma ^{{p_1},{q_1}}\) (1) are obtained, and the exact value of the modulus is found in the Hilbert case.