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In this article, we present new gradient estimates for positive solutions to a class of non‐linear elliptic equations involving the f‐Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet–Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry–Émery Ricci curvature tensor. From these estimates, we derive amongst other things Harnack inequalities and general global constancy and Liouville‐type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed. [ABSTRACT FROM AUTHOR]

We obtain pointwise upper bounds on the derivatives of the heat kernel on Damek–Ricci spaces and we study the Lp‐boundedness of Littlewood–Paley–Stein operators. [ABSTRACT FROM AUTHOR]

We study global gradient estimates for the weak solution to elliptic systems with measurable coefficients from composite materials in a nonsmooth bounded domain. The principal coefficients are assumed to be merely measurable in one variable and have small bounded mean oscillation (BMO) seminorms in the other variables on each subdomain whose boundary satisfies the so‐called δ‐Reifenberg flat condition. Under these assumptions and based on our new geometric observation for disjoint Reifenberg domains in a previous study, we obtain global W1,p estimates. [ABSTRACT FROM AUTHOR]

Abstract: Under some mild assumptions on the Lévy measure and the symbol we obtain gradient estimates of Dirichlet heat kernels for pure‐jump isotropic unimodal Lévy processes in R d. [ABSTRACT FROM AUTHOR]

We obtain global regularity in generalized Morrey spaces for the gradient of the weak solutions to divergence form linear parabolic operators with measurable data. Assuming partial BMO smallness of the coefficients and Reifenberg flatness of the boundary of the underlying domain, we develop a Calderón-Zygmund type theory for such operators. Problems like the considered here arise in the modeling of composite materials and in the mechanics of membranes and films of simple nonhomogeneous materials which form a linear laminated medium. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we prove the sharp global Li-Yau type gradient estimates for positive solutions to doubly nonlinear diffusion equation(DNDE) on complete Riemannian manifolds with nonnegative Ricci curvature. As an application, one can obtain a parabolic Harnack inequality. In the second part, we obtain a Perelman-type entropy monotonicity formula for DNDE on compact Riemannian manifolds with nonnegative Ricci curvature. These results generalize some works of Ni (JGA 2004), Lu-Ni-Vázquez-Villani (JMPA 2009) and Kotschwar-Ni (Annales Scientifiques de l'École Normale Supérieure 2009). Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]