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This article studies a nonlinear parabolic equation on a complete weighted manifold where the metric and potential evolve under a super Perelman-Ricci flow. It derives elliptic gradient estimates of local and global types for the positive solutions and exploits some of their implications notably to a general Liouville type theorem, parabolic Harnack inequalities and classes of Hamilton type dimension-free gradient estimates. Some examples and special cases are discussed for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we show that, for doubling manifolds satisfiying the scaled Poincaré inequalities and $p\in (2,\infty )$ , the boundedness of the Riesz transform dΔ on L , is essentially equivalent to the fact that $H_{1,d}^{p}$ is equal the L closure of the set of L exact harmonic 1-forms. Here, $H_{1,d}^{p}$ is a Hardy space of exact 1 −forms, naturally associated with the Riesz transform, as defined by Auscher, McIntosh and Russ. [ABSTRACT FROM AUTHOR]

We derive gradient and energy estimates for critical points of the full supersymmetric sigma model and discuss several applications. [ABSTRACT FROM AUTHOR]

We obtain a global weighted L estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg flat, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the Morrey spaces which implies global Hölder continuity of the solution. [ABSTRACT FROM AUTHOR]

In this note, we concentrate on the sub-Laplace operator on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by n Brownian motions and their $\frac{n(n-1)}2$ Lévy area processes, which is the simple extension of the sub-Laplacian on the Heisenberg group ℍ. In order to study contraction properties of the associated heat kernel, we show that, as in the cases of the Heisenberg group and the three Brownian motions model, the restriction of the sub-Laplace operator acting on radial functions (see Definition 3.5) satisfies a positive Ricci curvature condition (more precisely a CD(0, ∞ ) inequality), see Theorem 4.5, whereas the operator itself does not satisfy any CD( r, ∞ ) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel. It can be seen a generalization of the paper (Qian, Bull Sci Math 135:262-278, ). [ABSTRACT FROM AUTHOR]

By utilizing the Poincaré inequality and representation formulae, it is shown that on the Heisenberg type group, ℍ(2 n, m), there exists a constant C > 0 such that [ABSTRACT FROM AUTHOR]

In this paper, we introduce some techniques of Bakry–Emery curvature operator to Ricci flow and prove the evolution equation for the Bakry–Emery scalar curvature. As its application, we can easily derive the Perelman’s entropy functional monotonicity formula. We also discuss some gradient estimates of Ricci curvature and L 2– estimates of scalar curvature. [ABSTRACT FROM AUTHOR]