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

Abstract

Rubin's generalized Minkowski--Funk transforms $M_t^{\hat{I}}\pm$ on the sphere ${\mathbb{S}}^n$ give rise, for irrational radii $t=\cos ({\hat{I}}^2\ddot{I}€)$, to a small denominator problem governed by the asymptotic behavior of their spectral multipliers. We show that for Lebesgue-almost every ${\hat{I}}^2$ the corresponding two-sine small divisor inequality has infinitely many solutions, and deduce that $(M_t^{\hat{I}}\pm {)}^{-1}$ is not bounded from ${\tilde{H}}^{s+\ddot{I}+1}({\mathbb{S}}^n)$ to $H^s({\mathbb{S}}^n)$ in the non-critical case $\ddot{I}\ne 0,1$. In the critical cases $\ddot{I}\in \{0,1\}$ we prove Rubin's Conjectures 4.4 and 4.7 on the failure of endpoint Sobolev regularity for the inverse transforms.