No abstract

No abstract

Abstract. In this paper, we investigate Dirichlet spaces D µ with superharmonic weights induced by positive Borel measures µ on the open unit disk. We establish the AlexanderTaylor-Ullman inequality for D µ spaces and we characterize the cases where equality occurs. We de ne a class of weighted Hardy spaces H µ via the balayage of the measure µ. We show that D µ is equal to H µ if and only if µ is a Carleson measure for D µ . As an application, we obtain the reproducing kernel of D µ when µ is an in nite sum of point mass measures. We consider the boundary behavior and inner-outer factorization of functions in D µ . We also characterize the boundedness and compactness of composition operators on D µ .

Abstract. We prove variants of Schwarz's lemma involving monotonicity properties of condenser capacity and inner radius. Also, we examine when a similar monotonicity property holds for the hyperbolic metric.

Abstract. We prove that any first order F 2 Mm 1 ,m 2 ,n 1 ,n 2 -natural operator transforming projectable general connections on an (m1, m2, n1, n2The aim of this paper is to describe all F 2 M m 1 ,m 2 ,n 1 ,n 2 -natural operators transforming projectable general connections on an (m 1 , m 2 , n 1 , n 2 )-dimensional fibred-fibred manifolds into general connections on the vertical prolongation V Y → M of p : Y → M . The similar problem for the case of fibred manifolds was solving in [7]. In the paper [1], authors described natural operators transforming connections on fibred manifoldsA fibred-fibred manifold is a fibred surjective submersion