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# A Remark On the Moser-Aubin Inequality For Axially Symmetric Functions On the Sphere

Published:1999-12-01
Printed: Dec 1999
• Alexander R. Pruss
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## Abstract

Let $\scr S_r$ be the collection of all axially symmetric functions $f$ in the Sobolev space $H^1(\Sph^2)$ such that $\int_{\Sph^2} x_ie^{2f(\mathbf{x})} \, d\omega(\mathbf{x})$ vanishes for $i=1,2,3$. We prove that $$\inf_{f\in \scr S_r} \frac12 \int_{\Sph^2} |\nabla f|^2 \, d\omega + 2\int_{\Sph^2} f \, d\omega- \log \int_{\Sph^2} e^{2f} \, d\omega > -\oo,$$ 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: Moser inequality, borderline Sobolev inequalities, axially symmetric functions
 MSC Classifications: 26D15 - Inequalities for sums, series and integrals 58G30 - unknown classification 58G30