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A Remark on the Moser-Aubin Inequality for Axially Symmetric Functions on the Sphere
Published online by Cambridge University Press: 20 November 2018
Abstract
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Let 
${{\mathcal{S}}_{r}}$ be the collection of all axially symmetric functions 
$f$ in the Sobolev space 
${{H}^{1}}\left( {{\mathbb{S}}^{2}} \right)$ such that 
$\int_{{{\mathbb{S}}^{2}}}{{{x}_{i}}{{e}^{2f\left( x \right)}}\,dw\left( \text{x} \right)}$ vanishes for 
$i\,=\,1,\,2,\,3$. We prove that
$$\underset{f\in {{\mathcal{S}}_{r}}}{\mathop \inf }\,\frac{1}{2}\int_{{{\mathbb{S}}^{2}}}{{{\left| \nabla f \right|}^{2}}\,dw\,+\,2\,\int_{{{\mathbb{S}}^{2}}}{f\,dw\,-\,\log \,\int_{{{\mathbb{S}}^{2}}}{{{e}^{2f}}\,dw\,>\,-\infty ,}}}$$
and that this infimum is attained. This complements recent work of Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang concerning the Moser-Aubin inequality.
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- Research Article
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- Copyright © Canadian Mathematical Society 1999
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