Harnack-type inequalities for nonlocal parabolic equations in the upper fast-diffusion range

We study non-negative weak solutions to a class of doubly nonlinear nonlocal parabolic equations of the form $$(u^q{)}_t+\mathcal{L}u=0,$$ where $\mathcal{L}$ is a nonlocal $p$-Laplace-type operator with a symmetric measurable kernel. We consider the fast-diffusion range $1<p<q+1$ and establish Harnack-type estimates in the upper fast-diffusion regime ${\tilde{p}}_c<p<q+1$, where ${\tilde{p}}_c$ is a critical exponent determined by the interplay between the temporal nonlinearity and the nonlocal diffusion. In particular, the estimates apply to the nonlocal parabolic $p$-Laplace equation and the nonlocal porous medium equation.