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 Mat. Zametki, 2019, Volume 105, Issue 4, paper published in the English version journal (Mi mz11273)

Papers published in the English version of the journal

Proximinality in Banach space valued grand Bochner-Lebesgue spaces with variable exponent

Haihua Wei, Jingshi Xu

School of Mathematics and Statistics, Hainan Normal University, Haikou, 571158 China

Abstract: Let $(A,\mathscr{A},\mu)$ be a $\sigma$-finite complete measure space and $p(\cdot)$ be a $\mu$-measurable function on $A$ which takes values in $(1,\infty).$ Let $Y$ be a subspace of a Banach space $X.$ Denote $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ and $\widetilde{L}^{p(\cdot),\varphi}(A, X)$ by grand Bochner-Lebesgue spaces with variable exponent $p(\cdot)$ whose functions take values in $Y$ and $X$ respectively. Firstly, we estimate the distance of $f$ from $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ when $f\in \widetilde{L}^{p(\cdot),\varphi}(A, X).$ Then we obtain that $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ is proximinal in $\widetilde{L}^{p(\cdot),\varphi}(A, X)$ if $Y$ is weakly $\mathcal{K}$-analytic and proximinal in $X.$ Finally, we establish the connection between the proximinality of $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ in $\widetilde{L}^{p(\cdot),\varphi}(A, X)$ and the proximinality of $L^1(A, Y)$ in $L^1(A, X).$

Keywords: Proximinality; Grand Bochner-Lebesgue spaces; variable exponent; Best approximation; weakly $\mathcal{K}$-analytic

 Funding Agency Grant Number Natural Science Foundation of Hainan Province 2018CXTD338 National Natural Science Foundation of China 1176102611761027 The research of the second author was supported by the Natural Science Foundation of Hainan Province (Grant No. 2018CXTD338) and the National Natural Science Foundation of China (Grant No. 11761026 and 11761027).

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English version:
Mathematical Notes, 2019, 105:4, 618–624

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Revised: 25.04.2016

Citation: Haihua Wei, Jingshi Xu, “Proximinality in Banach space valued grand Bochner-Lebesgue spaces with variable exponent”, Math. Notes, 105:4 (2019), 618–624

Citation in format AMSBIB
\Bibitem{WeiXu19} \by Haihua~Wei, Jingshi~Xu \paper Proximinality in Banach space valued grand Bochner-Lebesgue spaces with variable exponent \jour Math. Notes \yr 2019 \vol 105 \issue 4 \pages 618--624 \mathnet{http://mi.mathnet.ru/mz11273} \crossref{https://doi.org/10.1134/S0001434619030349} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000467561600034} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85065663210}