We consider reflexive Bergman spaces Ap(Ω) on polygonal domains Ω of the complex plane. With some restrictions to angles boundary Ω, we show that boundedness Toeplitz operator Tg:Ap(Ω)→Ap(Ω) with a positive symbol g is equivalent Berezin transform or times area measure being Carleson measure. The result also formulated for more general simply connected domains. main technical tool new weighted ...

Let H°°(Dn) denote the set of all bounded analytic functions defined on the polydisc D" of Cn. In this note, we give a sufficient condition for sequences of points in £>" to be interpolating sequences for H°°(Dn). We also discuss some conditions for interpolation of general domains. Let G(X) be the set of all bounded continuous functions on a compact set X and let A C C(X) be a uniform algebra ...

Let \(\mu\) be a Beltrami coefficient on the unit disk, which is compatible with finitely generated Fuchsian group \(G\) of second kind. In this paper we show that if \(\frac{|\mu|^{2}}{1-|z|^{2}}\,dx\,dy\) satisfies Carleson condition infinite boundary Dirichlet fundamental domain \(G\), then measure disk.

We show that a maximal inequality holds for the non-tangential maximal operator on Dirichlet spaces with harmonic weights on the open unit disc. We then investigate two notions of Carleson measures on these spaces and use the maximal inequality to give characterizations of the Carleson measures in terms of an associated capacity.

Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this paper is to show that the boundary Harnack principle and the Carleson estimate are eq...

In the present paper we consider an elliptic divergence form operator in Rn∖Rd with d<n−1 and prove that its Green function is almost affine, sense normalized difference between a sufficiently far away pole suitable affine at every scale satisfies Carleson measure estimate. The coefficients of can be very oscillatory, only need to satisfy some condition similar traditional quadratic condition.