Imbeddings of weighted Sobolev spaces (Q2712737)

Analogues of the Hardy-type inequalities, in which derivatives are replaced by differences, are obtained. More precisely, the authors characterize the weights \(v\) and \(w\) such that the inequality NEWLINE\[NEWLINE \int_0^a|f(x)|^pv(x) dx \leq C\Biggl(v(a)\int_0^a|f(x)|^p dx +\int_0^a\|\Delta_tf\|^p_{L^p(0,a-t)}w(t) dt\Biggr) NEWLINE\]NEWLINE (where \(a>0\), \(1\leq p<\infty\), \(\Delta_tf(x)=f(x+t)-f(x)\), \(\|g\|_{L^p(0,a-t)}=(\int_0^{a-t}|g(x)|^p dx)^{1/p}\) and \(C\) is a positive constant independent of \(f\)) holds. Various modifications of this inequality are also discussed. In fact, this paper is a survey of results which are presented here without proofs. NEWLINENEWLINENEWLINEFor more details see authors' papers [\textit{V. I. Burenkov} and \textit{W. D. Evans} [Dokl. Akad. Nauk 355, No. 5, 583-585 (1997; Zbl 0978.46015) and \textit{V. I. Burenkov, W. D. Evans} and \textit{M. L. Goldman} [J. Inequal. Appl. 1, No. 1, 1-10 (1997; Zbl 0886.26013)].NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].