Validity of a Li–Yau inequality for the fractional Laplacian

Determine whether a Li–Yau-type inequality holds for positive solutions of the fractional heat equation ∂_t u + (−Δ)^{β/2} u = 0 on (0,∞) × R^d with β ∈ (0,2), specifically an upper bound on (−Δ)^{β/2}(log u)(t,x) by a function of time analogous to the classical Li–Yau inequality.

Background

The paper surveys Li–Yau and Harnack inequalities in nonlocal settings and discusses difficulties that arise when moving from local diffusion to jump operators. In the classical setting, the Li–Yau inequality bounds −Δ(log u) by d/(2t).

For the nonlocal fractional Laplacian L = −(−Δ)^{β/2} with β ∈ (0,2), the existence of an analogous Li–Yau inequality was historically unclear and was identified in the literature as a major open problem. The notes later present a proof strategy via reduction to the heat kernel that establishes such an inequality, but the sentence below explicitly records the earlier open status as framed in Garofalo’s survey.