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

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.