Fractal Remez inequality on the sphere and observability of the heat equation

Abstract

This paper is concerned with Remez-type inequalities and their applications in observability inequality. Our aim is twofold. First, we establish the following fractal Remez's inequality on the unit sphere ${\mathbb{S}}^{n-1}$ \begin{align*} \sup_{\mathbb{S}^{n-1}} |p|\le C(M,N,n,δ)\sup_{M} |p|, \end{align*} where $M\subset {\mathbb{S}}^{n-1}$ ($n\ge 2$) is a fractal set of positive $(n-2+\delta )$-Hausdorff content for arbitrary $\delta \in (0,1)$, and $p$ is a spherical polynomial of degree at most $N\in {\mathbb{Z}}^{+}$. Second, building upon this fractal framework, we establish sharp observability inequalities for the heat equation on the sphere, again valid for all $\delta \in (0,1)$, which improve the result of Burq and Moyano [J. Eur. Math. Soc. (JEMS), 25 (4) (2023)] in the spherical setting. Furthermore, as an additional application, we prove a lower-dimensional observability inequality for the heat equation with super-quadratic potentials $V(x)=|x{|}^{2m}$ ($m\in {\mathbb{Z}}^{+},m\ge 2$) on the whole space ${\mathbb{R}}^n$.

26 pages