Weighted inequality of Hardy type for elliptic operators
The analogue of Hardy's weighted inequality in n-dimensional space is given. Let \(C^{\infty}_ 0(R^ n\setminus 0)\) be the space of all \(C^{\infty}\)-functions with compact support in \(R^ n\setminus 0\), \(P(D)=\sum_{| \alpha | =m}a_{\alpha}D^{\alpha}\) \((D=- i\partial /\partial x)\) be an elliptic operator with constant coefficients. Denote as \(\Lambda\) the set consisting of all integers in case of \(m\geq n\) and of all integers from the intervals (-\(\infty,m- n]\cup [0,\infty)\) in case of \(m<n.\) Theorem. Let \(\lambda\in R\setminus \Lambda\), \(p\in (1,\infty)\). Then there exists a positive constant \(C_{\lambda,p}\) such that for each \(\phi \in C^{\infty}_ 0(R^ n\setminus 0)\) \[ \sum_{| \alpha | \leq m}\| r^{| \alpha | -\lambda} D^{\alpha}\phi \|^*_ p\leq C_{\lambda,p}\| r^{m-\lambda} P(D)\phi \|^*_ p. \] For \(p=1\), \(p=\infty\) we have a similar estimate - the only difference consists in the fact that on the left-hand side we sum over \(| \alpha | \leq m-1\). \((\| f\|^*_ p=(\int | f|^ p(dr/r)d\omega)^{1/p}\), where \(\omega\) is the measure on \(S^{n-1}.)\)
\(C^{\infty }\)-functions with compact support
scientific article; zbMATH DE number 4086148