On the small denominator problem for generalized Minkowski--Funk transforms

TL;DR

This paper investigates the small denominator problem for generalized Minkowski--Funk transforms on spheres, showing that for almost all irrational radii, the inverse transforms are unbounded in certain Sobolev spaces and confirming conjectures about endpoint regularity failure.

Contribution

It establishes the spectral behavior of these transforms for irrational radii and proves Rubin's conjectures on the failure of endpoint Sobolev regularity in critical cases.

Findings

Infinitely many solutions to the small divisor inequality for almost every irrational radius.

Inverse transforms are unbounded between specific Sobolev spaces in the non-critical case.

Confirmed Rubin's conjectures on the failure of endpoint Sobolev regularity for inverse transforms.

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.