On the small denominator problem for generalized Minkowski--Funk transforms
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908765299671040 |
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| author | Han, Rui Rahimi, Yaghoub |
| author_facet | Han, Rui Rahimi, Yaghoub |
| contents | Rubin's generalized Minkowski--Funk transforms $M_t^α$ on the sphere $\mathbb{S}^n$ give rise, for irrational radii $t=\cos(βπ)$, to a small denominator problem governed by the asymptotic behavior of their spectral multipliers. We show that for Lebesgue-almost every $β$ the corresponding two-sine small divisor inequality has infinitely many solutions, and deduce that $(M_t^α)^{-1}$ is not bounded from $\tilde{H}^{s+ρ+1}(\mathbb{S}^n)$ to $H^s(\mathbb{S}^n)$ in the non-critical case $ρ\neq 0,1$. In the critical cases $ρ\in\{0,1\}$ we prove Rubin's Conjectures 4.4 and 4.7 on the failure of endpoint Sobolev regularity for the inverse transforms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09547 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the small denominator problem for generalized Minkowski--Funk transforms Han, Rui Rahimi, Yaghoub Classical Analysis and ODEs Number Theory Rubin's generalized Minkowski--Funk transforms $M_t^α$ on the sphere $\mathbb{S}^n$ give rise, for irrational radii $t=\cos(βπ)$, to a small denominator problem governed by the asymptotic behavior of their spectral multipliers. We show that for Lebesgue-almost every $β$ the corresponding two-sine small divisor inequality has infinitely many solutions, and deduce that $(M_t^α)^{-1}$ is not bounded from $\tilde{H}^{s+ρ+1}(\mathbb{S}^n)$ to $H^s(\mathbb{S}^n)$ in the non-critical case $ρ\neq 0,1$. In the critical cases $ρ\in\{0,1\}$ we prove Rubin's Conjectures 4.4 and 4.7 on the failure of endpoint Sobolev regularity for the inverse transforms. |
| title | On the small denominator problem for generalized Minkowski--Funk transforms |
| topic | Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2601.09547 |