Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Rui Han, Yaghoub Rahimi

TL;DR

The paper analyzes the small denominator problem for Rubin's generalized Minkowski--Funk transforms $M_t^\alpha$ on $\mathbb{S}^n$, linking the bounded invertibility of $M_t^\alpha$ to the asymptotics of spectral multipliers $m_t^{\alpha}(j)$. In the non-critical regime $\rho\neq0,1$, it proves that for almost every irrational $\beta$ the two-sine Diophantine expression $F(\beta,j)$ has infinitely many small values, implying $(M_t^\alpha)^{-1}$ is unbounded between the Sobolev spaces $\tilde{H}^{s+\rho+1}$ and $H^s$. In the critical regime $\rho\in\{0,1\}$, it resolves Rubin's endpoint conjectures (Conjectures 4.4 and 4.7), showing failure of endpoint Sobolev regularity for the inverse. The main methodological advance is a two-parameter moving-target Khintchine-type theorem, together with a key moving-target Diophantine lemma, which translate spectral small-denominator phenomena into metric Diophantine statements on the circle and its arithmetic progressions.

Abstract

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.

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

TL;DR

The paper analyzes the small denominator problem for Rubin's generalized Minkowski--Funk transforms on , linking the bounded invertibility of to the asymptotics of spectral multipliers . In the non-critical regime , it proves that for almost every irrational the two-sine Diophantine expression has infinitely many small values, implying is unbounded between the Sobolev spaces and . In the critical regime , it resolves Rubin's endpoint conjectures (Conjectures 4.4 and 4.7), showing failure of endpoint Sobolev regularity for the inverse. The main methodological advance is a two-parameter moving-target Khintchine-type theorem, together with a key moving-target Diophantine lemma, which translate spectral small-denominator phenomena into metric Diophantine statements on the circle and its arithmetic progressions.

Abstract

Rubin's generalized Minkowski--Funk transforms on the sphere give rise, for irrational radii , to a small denominator problem governed by the asymptotic behavior of their spectral multipliers. We show that for Lebesgue-almost every the corresponding two-sine small divisor inequality has infinitely many solutions, and deduce that is not bounded from to in the non-critical case . In the critical cases we prove Rubin's Conjectures 4.4 and 4.7 on the failure of endpoint Sobolev regularity for the inverse transforms.
Paper Structure (7 sections, 16 theorems, 127 equations)

This paper contains 7 sections, 16 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Metric Diophantine input and perturbed approximation.
  3. Rubin's transforms, small denominators, and bounded inversion
  4. The non-critical case $\rho\neq 0,1$
  5. The critical case $\rho\in \{0,1\}$ and Rubin's Conjectures 4.4 and 4.7
  6. Proof of Theorem \ref{['thm:main_sd']}
  7. Khintchine's theorem for two-parameter moving targets

Key Result

Proposition 2.1

Let $t = \cos\theta$, $\theta = \beta\pi$ with $0<\beta<1$, $\beta\neq 1/2$, and let $\rho = \alpha + (n-1)/2 \notin \{0,1\}$. Then there exists a constant $c_\beta \neq 0$ (depending on $\alpha$, $n$ and $\beta$ but not on $j$) and real parameters such that as $j\to\infty$.

Theorems & Definitions (28)

  • Proposition 2.1: Rubin's asymptotics, truncated
  • Remark 2.2
  • Theorem 2.3: Rubin Rubin00
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8: Rubin Rubin00
  • Conjecture : Rubin Rubin00
  • Conjecture : Rubin Rubin00
  • Theorem 2.9
  • ...and 18 more