Generalized weighted Sobolev spaces (Q2781665)

Let \(w=\left( w_{0},...,w_{k}\right) \) be a vector weight function on \(\Omega\subset\mathbb{R}\) and \(\mu=\left( \mu_{0},...,\mu_{k}\right) \) a vector measure that is \(p\)-admissible (a definition is given in the paper) for \(w\) on \(\Omega\). The authors consider Sobolev spaces of the type \(W^{k,p}\left( \Omega,\mu\right) :=\{f:\overline{\Omega}\rightarrow\mathbb{C}\mid f^{\left( j\right) }\in AC_{loc}\left( \Omega^{\left( j\right) }\right)\) for \(j=0,1,...,k-1\) and \(\left\|f^{\left( j\right) }\right\|_{L^{p}\left( \overline{\Omega},\mu_{j}\right) }<\infty\) for \(j=\overline{0,k}\}\) equipped with the seminorms NEWLINE\[NEWLINE\left\|f\right\|_{W^{k,p}\left( \overline{\Omega},\mu\right) }:=\bigg( \sum\limits_{j=0} ^{k}\left\|f^{\left( j\right) }\right\|_{L^{p}\left( \overline{\Omega },\mu_{j}\right) }^{p}\bigg) ^{1/p},\quad\text{for }1\leq p<\inftyNEWLINE\]NEWLINE and NEWLINE\[NEWLINE\left\|f\right\|_{W^{k,\infty}\left( \overline{\Omega} ,\mu\right) }:=\max\limits_{j}\left\|f^{\left( j\right) }\right\|_{L^{\infty}\left( \overline{\Omega},\mu_{j}\right) }.NEWLINE\]NEWLINE Such spaces arise in the study of orthogonal polynomials on \(\mathbb{R}.\)NEWLINENEWLINENEWLINEThe authors study conditions under which \(W^{k,p}\left( \overline{\Omega },\mu\right) \) is complete and conditions on the measure \(\mu\) in order for \(C_{c}^{\infty}\left(\mathbb{R}\right)\) or \(C^{\infty}\left(\mathbb{R}\right) \) to be dense in \(W^{k,p}\left( \left[ a,b\right] ,\mu\right) \).NEWLINENEWLINENEWLINEThey also prove some inequalities that generalize classical results in Sobolev spaces with respect to Lebesgue measure.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00015].