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

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,\delta)\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 $\mathbb{R}^n$.