In this paper we get sharp conditions on a weight $v$ which allow us to obtain some weighted inequalities for a local Hardy-Littlewood Maximal operator defined on an open set in the Euclidean $n$-space. This result is applied to assure a pointwise convergence of the Laguerre heat-diffusion semigroup $u(x, t) = (T(t) f)(x)$ to $f$ when $t$ tends to zero for all functions $f$ in $L^{p}(v(x)dx)$ for $p$ greater than or equal to 1 and a weight $v$. In proving this we obtain weighted inequalities for the maximal operator associated to the Laguerre diffusion semigroup of the Laguerre differential operator of order greater than or equal to 0. Finally, as a by-product, we obtain weighted inequalities for the Riesz-Laguerre operators.

Let $X$ be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space $X^*,$ and let $K$ be a nonempty, closed and convex subset of $X$ with $0$ in its interior. Let $T$ be maximal monotone and $S$ a possibly unbounded pseudomonotone, or finitely continuous generalized pseudomonotone, or regular generalized pseudomonotone operator with domain $K$. Let $\phi$ be a proper, convex and lower semicontinuous function. New results are given concerning the solvability of perturbed variational inequalities involving the operator $T+S$ and the function $\phi$. The associated range results for nonlinear operators are also given, as well asextensions and/or improvements of known results of Kenmochi, Le, Browder, Browder and Hess, De Figueiredo, Zhou, and others.

To an ergodic, essentially free and measure-preserving action of a non-amenable Baumslag-Solitar group on a standard probability space, a flow is associated. The isomorphism class of the flow is shown to be an invariant of such actions of Baumslag-Solitar groups under weak orbit equivalence. Results on groups which are measure equivalent to Baumslag-Solitar groups are also provided.

In this note, we prove Cheng-Yau type local gradient estimate for harmonic functions on Alexandrov spaces with Ricci curvature bounded below. We adopt a refined version of Moser's iteration which is based on Zhang-Zhu's Bochner type formula in "Yau’s gradient estimates on Alexandrov spaces." Our result improves the previous one of Zhang-Zhu in the case of negative Ricci lower bound.

Let $L$ be a nonnegative self-adjoint operator satisfying Davies-Gaffney estimates on $L^2(X)$, where $X$ is a metric space. In this paper, we introduce and develop a new function space VMO$_{L}(X)$ of vanishing mean oscillation type associated to $L$. We then prove that the dual of VMO$_{L}(X)$ is the Hardy space $H_L(X)$ which was investigated in \cite{HLMMY}. Some characterizations of VMO$_{L}(X)$ are also established.

We obtain an asymptotic estimate of the Green function of a random walk on $\boldsymbol{Z}^2$ having zero mean and killed when it exits from the upper half plane. A little more than the second moment condition is assumed. The estimate obtained is used to derive an exact asymptotic form of the hitting distribution of the lower half plane of the walk. The higher dimensional walks are dealt with in the same way.