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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

Alexander R. Pruss
Alexander R. Pruss*
Affiliation:
Department of Philosophy, University of Pittsburgh Pittsburgh, Pennsylvania 15260, U.S.A., e-mail: [email protected]
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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.

Keywords

26D1558G30Moser inequalityborderline Sobolev inequalitiesaxially symmetric functions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 4 , 01 December 1999 , pp. 478 - 485
Copyright
Copyright © Canadian Mathematical Society 1999

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