$L^p$-Logvinenko-Sereda sets and $L^p$-Carleson measures on compact manifolds

TL;DR

This paper extends the characterization of $L^p$-Logvinenko-Sereda sets and $L^p$-Carleson measures to all $p$ on compact manifolds, including eigenfunctions on spheres, advancing understanding of geometric conditions for these sets.

Contribution

It provides a comprehensive characterization of $L^p$-Logvinenko-Sereda sets and $L^p$-Carleson measures on compact manifolds for all $p$, including new results for eigenfunctions on spheres.

Findings

Complete characterization on the standard sphere for $p > 2m/(m-1)$.

Conjecture and evidence for geometric control condition for $p < 2m/(m-1)$.

Progress on an open problem by Ortega-Cerdà and Pridhnani.

Abstract

Marzo and Ortega-Cerd\`a gave geometric characterizations for $L^p$-Logvinenko-Sereda sets on the standard sphere for all $1\le p<\infty$. Later, Ortega-Cerd\`a and Pridhnani further investigated $L^2$-Logvinenko-Sereda sets and $L^2$-Carleson measures on compact manifolds without boundary. In this paper, we characterize $L^p$-Logvinenko-Sereda sets and $L^p$-Carleson measures on compact manifolds with or without boundary for all $1<p<\infty$. Furthermore, we investigate $L^p$-Logvinenko-Sereda sets and $L^p$-Carleson measures for eigenfunctions on compact manifolds without boundary, and we completely characterize them on the standard sphere $S^m$ for $p > \frac{2m}{m-1}$. For the range $p < \frac{2m}{m-1}$, we conjecture that $L^p$-Logvinenko-Sereda sets for eigenfunctions on the standard sphere $S^m$ are characterized by the tubular geometric control condition and we provide some evidence. These results provide new progress on an open problem raised by Ortega-Cerd\`a and Pridhnani.