Stability of sharp Fourier restriction to spheres
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866912069572362240 |
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| author | Carneiro, Emanuel Negro, Giuseppe Silva, Diogo Oliveira e |
| author_facet | Carneiro, Emanuel Negro, Giuseppe Silva, Diogo Oliveira e |
| contents | In dimensions $d \in \{3,4,5,6,7\}$, we prove that the constant functions on the unit sphere $\mathbb{S}^{d-1}\subset \mathbb{R}^d$ maximize the weighted adjoint Fourier restriction inequality $$ \left| \int_{\mathbb{R}^d} |\widehat{fσ}(x)|^4\,\big(1 + g(x)\big)\,d x\right|^{1/4} \leq {\bf C} \, \|f\|_{L^2(\mathbb{S}^{d-1})}\,,$$ where $σ$ is the surface measure on $\mathbb{S}^{d-1}$, for a suitable class of bounded perturbations $g:\mathbb{R}^d \to \mathbb{C}$. In such cases we also fully classify the complex-valued maximizers of the inequality. In the unperturbed setting ($g = {\bf 0}$), this was established by Foschi ($d=3$) and by the first and third authors ($d \in \{4,5,6,7\}$) in 2015. Our methods also yield a new sharp adjoint restriction inequality on $\mathbb S^7\subset \mathbb R^8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_03412 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Stability of sharp Fourier restriction to spheres Carneiro, Emanuel Negro, Giuseppe Silva, Diogo Oliveira e Classical Analysis and ODEs Analysis of PDEs In dimensions $d \in \{3,4,5,6,7\}$, we prove that the constant functions on the unit sphere $\mathbb{S}^{d-1}\subset \mathbb{R}^d$ maximize the weighted adjoint Fourier restriction inequality $$ \left| \int_{\mathbb{R}^d} |\widehat{fσ}(x)|^4\,\big(1 + g(x)\big)\,d x\right|^{1/4} \leq {\bf C} \, \|f\|_{L^2(\mathbb{S}^{d-1})}\,,$$ where $σ$ is the surface measure on $\mathbb{S}^{d-1}$, for a suitable class of bounded perturbations $g:\mathbb{R}^d \to \mathbb{C}$. In such cases we also fully classify the complex-valued maximizers of the inequality. In the unperturbed setting ($g = {\bf 0}$), this was established by Foschi ($d=3$) and by the first and third authors ($d \in \{4,5,6,7\}$) in 2015. Our methods also yield a new sharp adjoint restriction inequality on $\mathbb S^7\subset \mathbb R^8$. |
| title | Stability of sharp Fourier restriction to spheres |
| topic | Classical Analysis and ODEs Analysis of PDEs |
| url | https://arxiv.org/abs/2108.03412 |