| Abstract: |
| In this talk, I will provide a brief overview of the history of Onofri`s inequality for the unit sphere, highlighting its connection with Trudinger-Moser type inequalities on Euclidean bounded domains. I will focus on an $N$-dimensional Euclidean version of Onofri`s inequality, proved by del Pino and Dolbeault, for smooth compactly supported functions in $\mathbb{R}^N$, with $N \geq 2$. I will prove that this inequality can be extended to a suitable weighted Sobolev space and, although there is no clear connection with standard Sobolev spaces on $\mathbb{S}^N$ via stereographic projection, I will show that it is equivalent to the logarithmic Moser-Trudinger inequality with sharp constant, obtained by Carleson and Chang for balls in $\mathbb{R}^N$. These results are part of a joint work with N. Borgia and S. Cingolani. |
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