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Edwards' Theorem establishes duality between a convex cone in the space of continuous functions on a compact space and the set of representing or Jensen measures for this cone. In this paper we prove non-compact versions of this theorem.

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 .

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Abstract. In this paper, we show that the Möbius invariant function space Q p can be generated by variant Dirichlet type spaces D µ, p induced by nite positive Borel measures µ on the open unit disk. A criterion for the equality between the space D µ, p and the usual Dirichlet type space D p is given. We obtain a su cient condition to construct di erent D µ, p spaces and we provide examples. We establish decomposition theorems for D µ, p spaces, and prove that the non-Hilbert space Q p is equal to the intersection of Hilbert spaces D µ, p . As an application of the relation between Q p and D µ, p spaces, we also obtain that there exist di erent D µ, p spaces; this is a trick to prove the existence without constructing examples.