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Smoothness of solutions of a convolution equation of restricted-type on the sphere
Classical Analysis and ODEs · Submitted 23 Sep 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that any such solution belongs to the class $C^\infty(\mathbb{S}^{d-1})$.
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The abstract, sentence by sentence
Let $\mathbb{S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb{R}^d$, $d\geq 2$, equipped with surface measure $\sigma_{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation \[ a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \] where $a\in C^\infty(\mathbb{S}^{d-1})$, $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb{S}^{d-1})$. We prove that any such solution belongs to the class $C^\infty(\mathbb{S}^{d-1})$. In particular, we show that all critical points associated to the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb{S}^{d-1}$ are $C^\infty$-smooth. This extends previous work of Christ & Shao to arbitrary dimensions and general even exponents, and plays a key role in a companion paper.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\mathbb{S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb{R}^d$, $d\geq 2$, equipped with surface measure $\sigma_{d-1}$. | background 1.00 | 0.07 | 0.01 | asserted 0.88 |
| s1 | An instance of our main result concerns the regularity of solutions of the convolution equation \[ a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \] where $a\in C^\infty(\mathbb{S}^{d-1})$, $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb{S}^{d-1})$. | contribution 0.54 | 0.35 | 0.08 | asserted 0.85 |
| s2 | We prove that any such solution belongs to the class $C^\infty(\mathbb{S}^{d-1})$. | result 1.00 | 0.96 | 0.79 | asserted 1.00 |
| s3 | In particular, we show that all critical points associated to the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb{S}^{d-1}$ are $C^\infty$-smooth. | result 0.58 | 0.93 | 0.12 | asserted 1.00 |
| s4 | This extends previous work of Christ & Shao to arbitrary dimensions and general even exponents, and plays a key role in a companion paper. | interpretation 0.41 | 0.43 | 0.04 | asserted 1.00 |
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Original abstract
Let $\mathbb{S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb{R}^d$, $d\geq 2$, equipped with surface measure $\sigma_{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation \[ a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \] where $a\in C^\infty(\mathbb{S}^{d-1})$, $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb{S}^{d-1})$. We prove that any such solution belongs to the class $C^\infty(\mathbb{S}^{d-1})$. In particular, we show that all critical points associated to the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb{S}^{d-1}$ are $C^\infty$-smooth. This extends previous work of Christ & Shao to arbitrary dimensions and general even exponents, and plays a key role in a companion paper.
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1 arXiv paper cite this in 6 sentences
Papers citing it · 5 sentences without a confident label
Cites it as support 1 paper
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Formula [ref] and [ref] together imply that the convolution $σ_d-1^∗m$ defines a continuous function on $^d$, provided $d=2$ and $m≥5$, or $d≥3$ and $m≥3$.
Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres · Spherical convolutions · 0.96
What it cites · 1 sentence without a confident label
Contests 1 paper
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Moreover, the case $(d,m)=(2,4)$ of Theorem [ref] completes the proof of the main result in [ref] , where the following issue was detected: in [ref] , the first (unnumbered) displayed equation on p. 9 seems to be incorrect.
On smoothness of extremizers of the Tomas-Stein inequality for $S^1$ · Introduction · contest 0.86
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