The limit of the optimal Li--Yau constant for the fractional heat equation

Source abstract

For the fractional heat equation $\partial_t u + (-Δ)^{β/2}u=0$ on $\mathbb{R}^d$ with $β\in(0,2)$, Weber and Zacher introduced the optimal constant $C_{\rm LY}(β,d)$ in the Li--Yau type estimate $$(-Δ)^{β/2}\log u(t,\cdot)\leq \frac {C_{\rm LY}(β,d)}{t},\quad t>0.$$ They asked whether $C_{\rm LY}(β,d)$ converges to the classical Li--Yau constant $d/2$ as $β\uparrow2$. In this note we prove that the answer is affirmative. More precisely, for every fixed $d\geq1$ there exists a constant $K_d>0$ such that $$\frac{d}β\leq C_{\rm LY}(β,d)\leq\frac{d}β+K_d(2-β)\log\frac{e}{2-β},\quad 1\leqβ<2.$$ Consequently, $\lim_{β\uparrow2}C_{\rm LY}(β,d)=d/2$. The proof has two main ingredients. First, the Bochner subordination formula, together with a concavity argument, shows that the supremum in Weber--Zacher's formula for $C_{\rm LY}(β,d)$ is attained at the origin. Second, the factor $2-β$ is derived from a lower bound for the fractional heat kernel profile in the large-distance regime, where the Lévy--Itô decomposition plays an important role.