We study the Krylov-Safonov type Holder estimate and Harnack inequality for viscosity solutions satisfying a uniformly parabolic equation only where the gradient with respect to the space-time variables is large.
In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre cite{CS1} are extended to those for the integro-differential operators associated with symmetric, regularly vary
ing kernels at zero. In particular, we obtain the uniform Harnack inequality and Holder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels $K_{sigma, beta}$ satisfying $$ K_{sigma,beta}(y)asymp frac{ 2-sigma}{|y|^{n+sigma}}left( logfrac{2}{|y|^2}right)^{beta(2-sigma)}quad mbox{near zero} $$ with respect to $sigmain(0,2)$ close to $2$ (for a given $betainmathbb R$), where the regularity estimates do not blow up as the order $ sigmain(0,2)$ tends to $2.$
In this paper, we obtain a uniform $W^{2,varepsilon}$-estimate of solutions to the fully nonlinear uniformly elliptic equations on Riemannian manifolds with a lower bound of sectional curvature using the ABP method.
We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabre proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional c
urvature of the underlying manifold is nonnegative. Later, Kim improved Cabres result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.
