A remark on the Hardy inequalities
Let \(X^ \ell= X^ \ell (\mathbb{R}^ d)\) (\(d\) even, \(2\ell\geq d\)) be the Hilbert space of all functions satisfying \[ \| u\|_{X^ \ell}= \int_{\mathbb{R}^ d} \Biggl( \bigl| \nabla^ \ell u\bigr|^ 2+ {{| u|^ 2} \over {| x|^{2\ell} (1+\log^ 2| x|)}} \Biggr) dx<\infty. \] Denote by \(E= E(d,\ell)\) the set of all non-negative \(k\) such that \(k=\ell- 2j- d/2\) for some \(j=0,1,\dots\;\). For \(u\in X^ \ell (\mathbb{R}^ d)\) put \[ u_ E= \sum_{k\in E} \sum_{\nu=1}^{m(d,k)} R_{k\nu} (r)\Phi_{k \nu} (\omega), \] where \((r,\omega)\) stands for the spherical coordinates in \(\mathbb{R}^ d\), \(P(d,k)\), \(k=0,1,\dots\), for the space of all spherical harmonics of degree \(k\), \(m(d,k):=\dim P(d,k)\), \(\{\Phi_{k\nu} (\omega)\}\), \(\nu=1,\dots, m(d,k)\), is an orthonormal (in \(L_ 2 (S^{d-1})\)) basis in \(P(d,k)\) and \[ R_{k\nu} (r)= \int_{S^{d-1}} u(r,\omega) \overline {\Phi_{k\nu} (\omega)} d\omega. \] Using an identity for \(\int_{\mathbb{R}^ d} | \nabla^ \ell u|^ 2 dx\) given by \textit{Yu. V. Egorov} and \textit{V. A. Kondrat'ev} [Mat. Sb. 181, No. 2, 147-166 (1990; Zbl 0734.35063)] the author proves that there are positive constants \(C_ i= C_ i(\ell,d)\), \(i=1,2\), such that \[ \int_{\mathbb{R}^ d} | \nabla^ \ell u|^ 2 d\geq C_ 1\int_{\mathbb{R}^ d} {{| u-u_ E|^ 2} \over {| x|^{2\ell}}} dx+C_ 2 \int_{\mathbb{R}^ d} {{| u_ E|^ 2} \over {| x|^ 2 \log^ 2| x|}} dx \] for all \(u\in X^ \ell (\mathbb{R}^ d)\) satisfying \(R_{k\nu} (1)=0\) with \(\nu= 1,\dots, m(d,k)\) and \(k\in E\).
scientific article; zbMATH DE number 604797