An electrical resistance network (ERN) is a weighted graph (G, μ). The edges are weighted by a function μ and interpreted as resistors of possibly varying strengths. The effective resistance metric R(x, y) is the natural notion of distance between two vertices x, y in the ERN. We discuss how R can be formulated in terms of an energy form E defined for functions on the vertices, a dissipation fo...

‎by considering a degenerate $p(x)-$laplacian equation‎, ‎a generalized compact embedding in weighted variable‎ ‎exponent sobolev space is presented‎. ‎multiplicity of positive solutions are discussed by applying fibering map approach for the‎ ‎corresponding nehari manifold‎.

The common topic of this thesis is boundedness of integral and supremal operators between function spaces with weights. The results of this work have the form of characterizations of validity of weighted operator inequalities for appropriate cones of functions. The outcome can be divided into three categories according to the particular type of studied operators and function spaces. The first p...

For spaces of analytic functions defined on an open set in $\mathbb{C}^n$ that satisfy certain nice properties, we show operators preserve shift-cyclic are necessarily weighted composition operators. Examples for which this result holds true consist the Hardy space $H^p(\mathbb{D}^n) \, (0 < p \infty)$, Drury-Arveson $\mathcal{H}^2_n$, and Dirichlet-type $\mathcal{D}_{\alpha} (\alpha \in \mathb...

H ·H := ̆ h = fg : f, g ∈ H ̄ = H ←↩ H is the product space of H2, by inner/outer factorization and Cauchy-Schwarz inequality. It is interesting, then, to find the dual space of H1. C. Fefferman [7] proved that, under the H2 paring (with some care), (H2 ·H2)∗ = (H1)∗ = BMO∩H(D) is the space of the analytic functions with bounded mean oscillation. The definition of BMO, born out of a problem in el...

In this paper we study the Hilbert space of analytic functions with finite Dirichlet integral in a connected open set C2 in the complex plane. We show that every such function can be represented as a quotient of two bounded analytic functions, each of which has a finite Dirichlet integral. This has several consequences for the structure of invariant subspaces of the algebra of multiplication op...

Consider a set of continuous maps from the interval [0, 1] to a domain in R. Although the topological boundary of this set in the path space is not smooth in general, by using the theory of functions of bounded variation (BV functions) on the Wiener space and the theory of Dirichlet forms, we can discuss the existence of the surface measure and the Skorokhod representation of the reflecting Orn...