Vector-valued functions integrable with respect to bilinear maps
type: Item
instance of: scholarly article
title: Vector-valued functions integrable with respect to bilinear maps
zbMATH Open document ID: 1171.42010
author: Oscar Blasco
author: J. M. Calabuig
published in: Taiwanese Journal of Mathematics
publication date: 2009-06-23T00:00:00Z
review text: Let \((\Omega,\Sigma,\mu)\) be a \(\sigma\)-finite complete measure space, \(1\leq p<\infty\), and let \(X,Y,Z\) be Banach spaces. Suppose \({\mathcal B}:X\times Y\to Z\) is a bounded bilinear map. It is introduced the notion of an \(X\)-valued function \(p\)-integrable with respect to \({\mathcal B}\): a function \(f:\Omega\to X\) is such whenever \(\sup\left\{\int_\Omega\|({\mathcal B}f(w),y)\|^p d\mu(w):\|y\|=1\right\}<\infty\). The spaces of functions integrable with respect to bilinear maps arising from the Hölder and the Young inequalities are identified. These results are applied to establish conditions on \(X\)-valued kernels \(k\) for the boundedness of integral operators \(T_k^{\mathcal B}: L^p(\Omega',Y)\to L^p(\Omega,Z)\) given by \(T_k^{\mathcal B}(g)(w)=\int_{\Omega'}{\mathcal B}(k(w,w'),g(w'))d\mu'(w')\), where \((\Omega',\Sigma',d\mu')\) is also a \(\sigma\)-finite complete measure space.
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 5567837
zbMATH Keywords: Vector-valued functions
zbMATH Keywords: bilinear map
zbMATH Keywords: Hölder inequality
zbMATH Keywords: Young inequality
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500405186
OpenAlex ID: W1554987742
DOI: 10.11650/TWJM/1500405186
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description: scientific article; zbMATH DE number 5567837