Concerning a type of Sobolev inequality on the sphere
type: Item
instance of: scholarly article
title: Concerning a type of Sobolev inequality on the sphere
zbMATH Open document ID: 0735.53029
DOI: 10.1007/BF01444699
author: Shigeo Kawai
published in: Mathematische Annalen
publication date: 1992-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164941
review text: 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.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4215949
zbMATH Keywords: best constants
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: conformal metric
zbMATH Keywords: prescribed scalar curvature
MaRDI profile type: Publication
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OpenAlex ID: W1971822264
description: scientific article; zbMATH DE number 4215949