 0774 Jean DOLBEAULT, Maria J. ESTEBAN, Gabriella TARANTELLO
 Weighted MoserTrudinger and HardySobolev inequalities]{A weighted
MoserTrudinger inequality and its relation to the CaffarelliKohn
Nirenberg inequalities in two space dimensions.
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Mar 29, 07

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Abstract. We first prove a weighted inequality of MoserTrudinger type
depending on a parameter, in the twodimensional Euclidean space. The
inequality holds for radial functions if the parameter is larger than
$1$. Without symmetry assumption, it holds if and only if the
parameter is in the interval $(1,0]$.
The inequality gives us some insight on the symmetry breaking
phenomenon for the extremal functions of the HardySobolev
inequality, as established by CaffarelliKohnNirenberg, in two space
dimensions. In fact, for suitable sets of parameters (asymptotically
sharp) we prove symmetry or symmetry breaking by means of a blowup
method. In this way, the weighted MoserTrudinger inequality appears
as a limit case of the HardySobolev inequality.
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