The second order Caffarelli-Kohn-Nirenberg identities and inequalities
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866917664005292032 |
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| author | Chen, Xiao-Ping Tang, Chun-Lei |
| author_facet | Chen, Xiao-Ping Tang, Chun-Lei |
| contents | This paper focuses on optimal constants and optimizers of the second order Caffarelli-Kohn-Nirenberg inequalities. Firstly, we aim to study optimal constants and optimizers for the following second order Caffarelli-Kohn-Nirenberg inequality in radial space: let $N\ge1$, $t\ge p>1$, \begin{equation}\label{0.1} \left(\int_{\mathbb{R}^N} \frac{|Δu|^p}{|x|^{pα}} \mathrm{d}x\right)^{\frac{1}{p}} \left[\int_{\mathbb{R}^N} \frac{\left|\nabla u\right|^{\frac{p(t-1)}{p-1}}} {|x|^{\frac{p(t-1)}{p-1}β}} \mathrm{d}x\right]^{\frac{p-1}{p}} \ge C(N,p,t,α,β) \int_{\mathbb{R}^N} \frac{\left|\nabla u\right|^t}{|x|^{tγ}} \mathrm{d}x. \end{equation} Secondly, we establish second order $L^p$-Caffarelli-Kohn-Nirenberg identities, and obtain optimal constants and optimizers of the second order $L^p$-Caffarelli-Kohn-Nirenberg inequalities (i.e., $p=t$ in \eqref{0.1}) in general space. Lastly, under some more general assumptions, we consider the optimal weighted second order Heisenberg Uncertainty Principles, which complements the recent work [``The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle'', 2022, arXiv:2102.01425].
This paper's main novelty lies in the fact that we research the optimal versions of the second order Caffarelli-Kohn-Nirenberg inequalities \eqref{0.1} in radial space or in general space, and also establish the second order $L^p$-Caffarelli-Kohn-Nirenberg identities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06898 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The second order Caffarelli-Kohn-Nirenberg identities and inequalities Chen, Xiao-Ping Tang, Chun-Lei Analysis of PDEs This paper focuses on optimal constants and optimizers of the second order Caffarelli-Kohn-Nirenberg inequalities. Firstly, we aim to study optimal constants and optimizers for the following second order Caffarelli-Kohn-Nirenberg inequality in radial space: let $N\ge1$, $t\ge p>1$, \begin{equation}\label{0.1} \left(\int_{\mathbb{R}^N} \frac{|Δu|^p}{|x|^{pα}} \mathrm{d}x\right)^{\frac{1}{p}} \left[\int_{\mathbb{R}^N} \frac{\left|\nabla u\right|^{\frac{p(t-1)}{p-1}}} {|x|^{\frac{p(t-1)}{p-1}β}} \mathrm{d}x\right]^{\frac{p-1}{p}} \ge C(N,p,t,α,β) \int_{\mathbb{R}^N} \frac{\left|\nabla u\right|^t}{|x|^{tγ}} \mathrm{d}x. \end{equation} Secondly, we establish second order $L^p$-Caffarelli-Kohn-Nirenberg identities, and obtain optimal constants and optimizers of the second order $L^p$-Caffarelli-Kohn-Nirenberg inequalities (i.e., $p=t$ in \eqref{0.1}) in general space. Lastly, under some more general assumptions, we consider the optimal weighted second order Heisenberg Uncertainty Principles, which complements the recent work [``The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle'', 2022, arXiv:2102.01425]. This paper's main novelty lies in the fact that we research the optimal versions of the second order Caffarelli-Kohn-Nirenberg inequalities \eqref{0.1} in radial space or in general space, and also establish the second order $L^p$-Caffarelli-Kohn-Nirenberg identities. |
| title | The second order Caffarelli-Kohn-Nirenberg identities and inequalities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.06898 |