Higher Hölder regularity for a subquadratic nonlocal parabolic equation

In this paper, we are concerned with the Hölder regularity for solutions of the nonlocal evolutionary equation $${\mathrm{\partial }}_{t}u+(-{\mathrm{\Delta }}_p{)}^{s}u=0.$$ Here, $(-{\mathrm{\Delta }}_p{)}^s$ is the fractional $p$-Laplacian, $0<s<1$ and $1<p<2$. We establish Hölder regularity with explicit Hölder exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained Hölder exponents are almost sharp. Our results complement the previous results for the superquadratic case when $p\ge 2$.