On the equivalence between an Onofri-type inequality by Del Pino-Dolbeault and the sharp logarithmic Moser-Trudinger inequality
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910843609808896 |
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| author | Borgia, Natalino Cingolani, Silvia Mancini, Gabriele |
| author_facet | Borgia, Natalino Cingolani, Silvia Mancini, Gabriele |
| contents | In this paper we consider the $N$-dimensional Euclidean Onofri inequality proved by del Pino and Dolbeault for smooth compactly supported functions in $\mathbb{R}^N$, $N \geq 2$. We extend the inequality to a suitable weighted Sobolev space, although no clear connection with standard Sobolev spaces on $\mathbb{S}^N$ through stereographic projection is present, except for the planar case. Moreover, in any dimension $N \geq 2$, we show that the Euclidean Onofri inequality is equivalent to the logarithmic Moser-Trudinger inequality with sharp constant proved by Carleson and Chang for balls in $\mathbb{R}^N$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_04335 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the equivalence between an Onofri-type inequality by Del Pino-Dolbeault and the sharp logarithmic Moser-Trudinger inequality Borgia, Natalino Cingolani, Silvia Mancini, Gabriele Analysis of PDEs 26D10, 46E30, 46E35, 58E35, 35J92 In this paper we consider the $N$-dimensional Euclidean Onofri inequality proved by del Pino and Dolbeault for smooth compactly supported functions in $\mathbb{R}^N$, $N \geq 2$. We extend the inequality to a suitable weighted Sobolev space, although no clear connection with standard Sobolev spaces on $\mathbb{S}^N$ through stereographic projection is present, except for the planar case. Moreover, in any dimension $N \geq 2$, we show that the Euclidean Onofri inequality is equivalent to the logarithmic Moser-Trudinger inequality with sharp constant proved by Carleson and Chang for balls in $\mathbb{R}^N$. |
| title | On the equivalence between an Onofri-type inequality by Del Pino-Dolbeault and the sharp logarithmic Moser-Trudinger inequality |
| topic | Analysis of PDEs 26D10, 46E30, 46E35, 58E35, 35J92 |
| url | https://arxiv.org/abs/2408.04335 |