In this paper, we are concerned with the Hölder regularity forsolutions of the nonlocal evolutionary equation ∂tu + (−p) su = 0. Here, (−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 ≥ 2.

We study the boundary behavior of solutions to fractional Laplacian. As the first result, the isolation of the first eigenvalue of the fractional Lane-Emden equation is proved in the bounded open sets with Wiener regular boundary. Then, a generalized Hopf's lemma and a global boundary Harnack inequality are proved for the fractional Laplacian.

We study local boundedness and Hölder continuity of a parabolic equation involving the fractional p-Laplacian of order s, with 0<s<1, 2≤p<∞, with a general right-hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right-hand side.