dorsal/arxiv
View SchemaOn the small denominator problem for generalized Minkowski--Funk transforms
| Authors | Rui Han, Yaghoub Rahimi |
|---|---|
| Categories | |
| ArXiv ID | 2601.09547vv1 |
| URL | https://arxiv.org/abs/2601.09547 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
Rubin's generalized Minkowski--Funk transforms $M_t^\alpha$ on the sphere $\mathbb{S}^n$ give rise, for irrational radii $t=\cos(\beta\pi)$, to a small denominator problem governed by the asymptotic behavior of their spectral multipliers. We show that for Lebesgue-almost every $\beta$ the corresponding two-sine small divisor inequality has infinitely many solutions, and deduce that $(M_t^\alpha)^{-1}$ is not bounded from $\tilde{H}^{s+\rho+1}(\mathbb{S}^n)$ to $H^s(\mathbb{S}^n)$ in the non-critical case $\rho\neq 0,1$. In the critical cases $\rho\in\{0,1\}$ we prove Rubin's Conjectures 4.4 and 4.7 on the failure of endpoint Sobolev regularity for the inverse transforms.
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"abstract": "Rubin\u0027s generalized Minkowski--Funk transforms $M_t^\\alpha$ on the sphere $\\mathbb{S}^n$ give rise, for irrational radii $t=\\cos(\\beta\\pi)$, to a small denominator problem governed by the asymptotic behavior of their spectral multipliers. We show that for Lebesgue-almost every $\\beta$ the corresponding two-sine small divisor inequality has infinitely many solutions, and deduce that $(M_t^\\alpha)^{-1}$ is not bounded from $\\tilde{H}^{s+\\rho+1}(\\mathbb{S}^n)$ to $H^s(\\mathbb{S}^n)$ in the non-critical case $\\rho\\neq 0,1$. In the critical cases $\\rho\\in\\{0,1\\}$ we prove Rubin\u0027s Conjectures 4.4 and 4.7 on the failure of endpoint Sobolev regularity for the inverse transforms.",
"arxiv_id": "2601.09547",
"authors": [
"Rui Han",
"Yaghoub Rahimi"
],
"categories": [
"math.CA",
"math.NT"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "On the small denominator problem for generalized Minkowski--Funk transforms",
"url": "https://arxiv.org/abs/2601.09547",
"version": "v1"
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