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 Trudy Inst. Mat. i Mekh. UrO RAN, 2017, Volume 23, Number 3, Pages 22–32 (Mi timm1434)

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

R. R. Akopyanab, M. S. Saidusajnovc

a Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
c Tajik National University, Dushanbe

Abstract: 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.

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.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-02705 Ministry of Education and Science of the Russian Federation ÍØ-9356.2016.1 Ural Federal University named after the First President of Russia B. N. Yeltsin 02.A03.21.0006 îò 27.08.2013

DOI: https://doi.org/10.21538/0134-4889-2017-23-3-22-32

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2018, 303, suppl. 1, 25–35

Bibliographic databases:

UDC: 517.977
MSC: 30E10, 47A58

Citation: R. R. Akopyan, M. S. Saidusajnov, “Three extremal problems in the Hardy and Bergman spaces of functions analytic in a disk”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 3, 2017, 22–32; Proc. Steklov Inst. Math. (Suppl.), 303, suppl. 1 (2018), 25–35

Citation in format AMSBIB
\Bibitem{AkoSai17} \by R.~R.~Akopyan, M.~S.~Saidusajnov \paper Three extremal problems in the Hardy and Bergman spaces of functions analytic in a disk \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2017 \vol 23 \issue 3 \pages 22--32 \mathnet{http://mi.mathnet.ru/timm1434} \crossref{https://doi.org/10.21538/0134-4889-2017-23-3-22-32} \elib{https://elibrary.ru/item.asp?id=29295247} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2018 \vol 303 \issue , suppl. 1 \pages 25--35 \crossref{https://doi.org/10.1134/S0081543818090031} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000453521100002} 

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