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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.02303 |
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| _version_ | 1866914773333966848 |
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| author | De Nitti, Nicola Glaudo, Federico König, Tobias |
| author_facet | De Nitti, Nicola Glaudo, Federico König, Tobias |
| contents | The fractional Caffarelli-Kohn-Nirenberg inequality states that $$ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))^2}{|x|^α|x-y|^{n+2s} |y|^α} \mathrm{d} x \, \mathrm{d} y
\geq Λ_{n, s, p, α,β}
\|u |x|^{-β}\|_{L^p}^2, $$ for $0<s<\min\{1, n/2\}$, $2<p<2^*_s$, and $α,β\in\mathbb R$ so that $β-α= s - n\big(\frac12 - \frac1p\big)$ and $-2s < α< \frac{n-2s}{2}$.
Continuing the program started in Ao et al. (2022), we establish the non-degeneracy and sharp quantitative stability of minimizers for $α\ge 0$. Furthermore, we show that minimizers remain symmetric when $α<0$ for $p$ very close to $2$.
Our results fit into the more ambitious goal of understanding the symmetry region of the minimizers of the fractional Caffarelli-Kohn-Nirenberg inequality.
We develop a general framework to deal with fractional inequalities in $\mathbb R^n$, striving to provide statements with a minimal set of assumptions. Along the way, we discover a Hardy-type inequality for a general class of radial weights that might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02303 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-degeneracy, stability and symmetry for the fractional Caffarelli-Kohn-Nirenberg inequality De Nitti, Nicola Glaudo, Federico König, Tobias Analysis of PDEs The fractional Caffarelli-Kohn-Nirenberg inequality states that $$ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))^2}{|x|^α|x-y|^{n+2s} |y|^α} \mathrm{d} x \, \mathrm{d} y \geq Λ_{n, s, p, α,β} \|u |x|^{-β}\|_{L^p}^2, $$ for $0<s<\min\{1, n/2\}$, $2<p<2^*_s$, and $α,β\in\mathbb R$ so that $β-α= s - n\big(\frac12 - \frac1p\big)$ and $-2s < α< \frac{n-2s}{2}$. Continuing the program started in Ao et al. (2022), we establish the non-degeneracy and sharp quantitative stability of minimizers for $α\ge 0$. Furthermore, we show that minimizers remain symmetric when $α<0$ for $p$ very close to $2$. Our results fit into the more ambitious goal of understanding the symmetry region of the minimizers of the fractional Caffarelli-Kohn-Nirenberg inequality. We develop a general framework to deal with fractional inequalities in $\mathbb R^n$, striving to provide statements with a minimal set of assumptions. Along the way, we discover a Hardy-type inequality for a general class of radial weights that might be of independent interest. |
| title | Non-degeneracy, stability and symmetry for the fractional Caffarelli-Kohn-Nirenberg inequality |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.02303 |