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
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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.
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
@article{10_4153_CMB_1999_055_8,
author = {Pruss, Alexander R.},
title = {A {Remark} on the {Moser-Aubin} {Inequality} for {Axially} {Symmetric} {Functions} on the {Sphere}},
journal = {Canadian mathematical bulletin},
pages = {478--485},
year = {1999},
volume = {42},
number = {4},
doi = {10.4153/CMB-1999-055-8},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-055-8/}
}
TY - JOUR AU - Pruss, Alexander R. TI - A Remark on the Moser-Aubin Inequality for Axially Symmetric Functions on the Sphere JO - Canadian mathematical bulletin PY - 1999 SP - 478 EP - 485 VL - 42 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-055-8/ DO - 10.4153/CMB-1999-055-8 ID - 10_4153_CMB_1999_055_8 ER -
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