A strong-form stability for a class of $L^p$ Caffarelli-Kohn-Nirenberg interpolation inequality
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917792697024512 |
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| author | Zhang, Yingfang Zou, Wenming |
| author_facet | Zhang, Yingfang Zou, Wenming |
| contents | We study the stability of a class of Caffarelli-Kohn-Nirenberg (CKN) interpolation inequality and establish a strong-form stability as following: \begin{equation*}
\inf_{v\in\mathcal{M}_{p,a,b}}\frac{ \|u-v\|_{H_b^p} \|u-v\|_{L^p_a}^{p-1} }{\|u\|_{H^p_b}\|u\|_{L^p_a}^{p-1}} \le Cδ_{p,a,b}(u)^{t}, \end{equation*} where $t=1$ for $p=2$ and $t=\frac{1}{p}$ for $p > 2$, and $δ_{p,a,b}(u)$ is deficit of the CKN. We also note that it is impossible to establish stability results for $\|\cdot\|_{H_b^p}$ or $\|\cdot\|_{L_a^p}$ separately. Moreover, we consider the second-order CKN inequalities and establish similar results for radial functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00777 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A strong-form stability for a class of $L^p$ Caffarelli-Kohn-Nirenberg interpolation inequality Zhang, Yingfang Zou, Wenming Analysis of PDEs Functional Analysis We study the stability of a class of Caffarelli-Kohn-Nirenberg (CKN) interpolation inequality and establish a strong-form stability as following: \begin{equation*} \inf_{v\in\mathcal{M}_{p,a,b}}\frac{ \|u-v\|_{H_b^p} \|u-v\|_{L^p_a}^{p-1} }{\|u\|_{H^p_b}\|u\|_{L^p_a}^{p-1}} \le Cδ_{p,a,b}(u)^{t}, \end{equation*} where $t=1$ for $p=2$ and $t=\frac{1}{p}$ for $p > 2$, and $δ_{p,a,b}(u)$ is deficit of the CKN. We also note that it is impossible to establish stability results for $\|\cdot\|_{H_b^p}$ or $\|\cdot\|_{L_a^p}$ separately. Moreover, we consider the second-order CKN inequalities and establish similar results for radial functions. |
| title | A strong-form stability for a class of $L^p$ Caffarelli-Kohn-Nirenberg interpolation inequality |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2410.00777 |