Concerning a type of Sobolev inequality on the sphere
This paper treats the so-called best constants in the Sobolev inequality. The following result is proved: There exists a constant \(a\) (\(0<a<1\)) such that \[ n(n-1)\{\int_{\hbox{avg}}| u|^{2n/(n-2)}\}^{(n- 2)/n}\leq a\{4(n-1)/(n-2)\}\int_{\hbox{avg}}|\nabla u|^ 2+n(n- 1)\int_{\hbox{avg}} u^ 2 \] for \(u\in W^{1,2}(S^ n)\) (\(n\geq3\)) which satisfy the equality \(\int_{\hbox{avg}}| u|^{2n/(n- 2)}x_ j=0\) (\(j=1,2,\dots,n+1\)), where \(x_ j\) are orthogonal coordinates in \(R^{n+1}\), and \(\int_{\hbox{avg}}\) denotes the integral average on \(S^ n\). This type of Sobolev inequality was proved for functions on \(S^ 2\) by S. Y. Chang and and P. C. Yang, and played an important role in the proof of the existence of a pointwise conformal metric on \(S^ 2\) with prescribed scalar curvature. The method of proof for higher dimensional case is different from that of S. Y. Chang and P. C. Yang, and it can also be applied to the 2-dimensional case.
scientific article; zbMATH DE number 4215949