On a set everywhere dense in a Lebesgue space on the real line
If \({\mathcal L}\) denotes the space of Lebesgue measurable functions on the real line \({\mathcal R}= (-\infty,\infty)\), if \(1\leq p<\infty\), and if \(E\) is a Lebesgue measurable subset of \({\mathcal R}\), let \(L^p(E)\) denote \(\{f\in{\mathcal L}:\|[f]\|_{p,E}= \| f\|_{p,E}< \infty\}\), where \(\| f\|_{p,E}= (\int_E|f(x)|^p\,dx)^{1/p}\). If \(1\leq p<\infty\), \((1/p)+ (1/p')= 1\), then it is known, as also indicated in the introductory section of this paper, that \[ \begin{aligned} & \sup\{(\| h\|_{p',E})^{-1}|\int fh\,dm: h\in L^{p'}(E)\}\\ &= \sup\Biggl\{(\| h\|_{p',E})^{-1} \int_E |fh|\,dm: h\in L^{p'}(E)\Biggr\}\\ &=\| f\|_{p,E}.\end{aligned} \] One of the main conclusion of this paper indicates that, if \(Y\) is a dense subspace of \(L^{p'}(E)\), then \[ \sup\Biggl\{(\| h\|_{p',E})^{-1} \int_E |fh|\,dm: h\in Y\Biggr\}=\| f\|_{p,E}. \] Other statements of the paper involve the space \(W^1_{p,v,E}\) consisting of locally differentiable functions with norm \(\| u\|_{p,E}+\| D(u)\|_{p,E}\), having a dense subspace consisting of continuously differentiable functions.
scientific article; zbMATH DE number 6682280