Geodesic
A Remark on the Moser-Aubin Inequality for Axially Symmetric Functions on the Sphere
Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 478-485

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1999-055-8
Mots-clés : 26D15, 58G30, Moser inequality, borderline Sobolev inequalities, axially symmetric functions 
Pruss, Alexander R. A Remark on the Moser-Aubin Inequality for Axially Symmetric Functions on the Sphere. Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 478-485. doi: 10.4153/CMB-1999-055-8
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