Borderline gradient continuity for the normalized p-parabolic operator

In this paper, we prove gradient continuity estimates for viscosity solutions to ΔNpu−ut=f in terms of the scaling critical L(n+2,1) norm of f, where ΔNp is the game theoretic normalized p−Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for u in terms of the modified parabolic Riesz potential Pfn+1 as defined in (2.8) below. Moreover, for f∈Lm with m>n+2, we also obtain Hölder continuity of the spatial gradient of the solution u, see Theorem 2.6 below. This improves the gradient Hölder continuity result in [3] which considers bounded f. Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in [9]. Moreover differently from that in [3], our approach is independent of the Ishii-Lions method which is crucially used in [3] to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.