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

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 $\rho \mathrm{\ne }0,1$. 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.