A Local Douglas formula for Higher Order Weighted Dirichlet-Type Integrals

We prove a local Douglas formula for higher order weighted Dirichlet-type integrals. With the help of this formula, we study the multiplier algebra of the associated higher order weighted Dirichlet-type spaces H-mu, induced by an m-tuple mu = (mu(1),..., mu(m)) of finite non-negative Borel measures on the unit circle. In particular, it is shown that any weighted Dirichlet-type space of order m, for m >= 3, forms an algebra under pointwise product. We also prove that every non-zero closed M-z-invariant subspace of H-mu, has codimension 1 property if m >= 3 or mu(2) is finitely supported. As another application of this local Douglas formula obtained in this article, it is shown that for any m >= 2, weighted Dirichlet-type space of order m does not coincide with any de Branges-Rovnyak space H(b) with equivalence of norms.