Three extremal problems in the Hardy and Bergman spaces of functions analytic in a disk

R. R. Akopyan ; M. S. Saidusajnov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 3, pp. 22-32

Résumé

Let a nonnegative measurable function $\gamma(\rho)$ be nonzero almost everywhere on $(0,1)$, and let the product $\rho\gamma(\rho)$ be summable on $(0,1)$. Denote by $\mathcal{B}=B^{p,q}_{\gamma}$, $1\leq p\le \infty$, $1\leq q \infty$, the space of functions $f$ analytic in the unit disk for which the function $M_p^q(f,\rho)\rho\gamma(\rho)$ is summable on $(0,1)$, where $M_p^q(f,\rho)$ is the $p$-mean of $f$ on the circle of radius $\rho$; this space is equipped with the norm $$ \|f\|_{B^{p,q}_{\gamma}}=\|M_p(f,\cdot)\|_{L^q_{\rho\gamma(\rho)}(0,1)}. $$ In the case $q=\infty$, the space $\mathcal{B}=B^{p,\infty}_{\gamma}$ is identified with the Hardy space $H^p$. Using an operator $L$ given by the equality $Lf(z)=\sum_{k=0}^\infty l_k c_k z^k$ on functions $f(z)=\sum_{k=0}^\infty c_k z^k$ analytic in the unit disk, we define the class $$ LB_\gamma^{p,q}(N):=\{f\colon \|Lf\|_{B_\gamma^{p,q}}\le N\},\quad N>0. $$ 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 $LB_\gamma^{p_1,q_1}(1)$ 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 $2\le p_{1}\le\infty$, $1\leq p_{2}\leq 2$, $1\leq p_{3}\leq 2$, $1\le q_1=q_2=q_3\le\infty$, and $q_s=2$ or $\infty$. (2) The best approximation of the operator $L$ by the set $\mathcal{L}(N)$, $N>0$, of linear bounded 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 $2\le p_{1}\le\infty$, $1\leq p_{2}\leq 2$, $2\leq p_{3}\leq \infty$, $1\le q_1=q_2=q_3\le\infty$, and $q_s=2$ or $\infty$. (3) Bounds for the modulus of continuity of the operator $L$ on the class $GB_\gamma^{p_3,q_3}(1)$ are obtained, and the exact value of the modulus is found in the Hilbert case.

Détail

Keywords: Hardy and Bergman spaces, best approximation of a class by a class, best approximation of an unbounded operator by bounded operators, modulus of continuity of an operator.

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