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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.19930 |
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| _version_ | 1866908506173472768 |
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| author | Ghosh, Monideep Karmakar, Debabrata |
| author_facet | Ghosh, Monideep Karmakar, Debabrata |
| contents | In this work, we focus on a recent variant of the Trudinger-Moser-Onofri inequality introduced by S. Y. Alice Chang and Changfeng Gui \cite{CG-2023}: \begin{align*} α\int_{\mathbb{S}^2}|\nabla_{\mathbb{S}^2}u|^2 {\rm d}ω+2 \int_{\mathbb{S}^2} u {\rm d}ω-\frac{1}{2}\ln\left[\left(\int_{\mathbb{s}^2}e^{2u}{\rm d}ω\right)^2-\sum_{i=1}^3\left(\int_{\mathbb{s}^2}ω_i e^{2u}{\rm d} ω\right)^2\right] \geq 0 \end{align*} holds on $H^1(\mathbb{S}^2)$ if and only if $α\geq \frac{2}{3}$. In this regime, the infimum is attained only by trivial functions when $α> \frac{2}{3},$ whereas for the critical value $α= \frac{2}{3}$ nontrivial extremals exist, and Chang-Gui further provided a complete classification of such solutions.
Building upon their result, we found a nice conformal invariance of the associated functional. Exploiting this invariance, we were able to characterize the full family of extremals in terms of conformal maps of $\mathbb{S}^2$ and, moreover, establish a sharp quantitative stability result in the gradient norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative stability for the conformally invariant Chang-Gui inequality on the exponentiation of functions on the sphere Ghosh, Monideep Karmakar, Debabrata Analysis of PDEs Functional Analysis In this work, we focus on a recent variant of the Trudinger-Moser-Onofri inequality introduced by S. Y. Alice Chang and Changfeng Gui \cite{CG-2023}: \begin{align*} α\int_{\mathbb{S}^2}|\nabla_{\mathbb{S}^2}u|^2 {\rm d}ω+2 \int_{\mathbb{S}^2} u {\rm d}ω-\frac{1}{2}\ln\left[\left(\int_{\mathbb{s}^2}e^{2u}{\rm d}ω\right)^2-\sum_{i=1}^3\left(\int_{\mathbb{s}^2}ω_i e^{2u}{\rm d} ω\right)^2\right] \geq 0 \end{align*} holds on $H^1(\mathbb{S}^2)$ if and only if $α\geq \frac{2}{3}$. In this regime, the infimum is attained only by trivial functions when $α> \frac{2}{3},$ whereas for the critical value $α= \frac{2}{3}$ nontrivial extremals exist, and Chang-Gui further provided a complete classification of such solutions. Building upon their result, we found a nice conformal invariance of the associated functional. Exploiting this invariance, we were able to characterize the full family of extremals in terms of conformal maps of $\mathbb{S}^2$ and, moreover, establish a sharp quantitative stability result in the gradient norm. |
| title | Quantitative stability for the conformally invariant Chang-Gui inequality on the exponentiation of functions on the sphere |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2508.19930 |