We give a method allowing the generalization of the description of trace spaces of certain classes of holomorphic functions on Carleson sequences to ®nite unions of Carleson sequences. We apply the result to di€erent classes of spaces of holomorphic functions such as Hardy classes and Bergman type spaces. 0. Introduction. Let D={z2C : jzj<1} be the unit disk and Hol(D) the space of holomorphic ...

In this article, we highlight the role of Carleson measures in elliptic boundary value prob5 lems, and discuss some recent results in this theory. The focus here is on the Dirichlet problem, with 6 measurable data, for second order elliptic operators in divergence form. We illustrate, through selected 7 examples, the various ways Carleson measures arise in characterizing those classes of operat...

Trudinger and Moser, interested in certain nonlinear problems in differential geometry, showed that if |∇u|q is integrable on a bounded domain in R with q ≥ n ≥ 2, then u is exponentially integrable there. Symmetrization reduces the problem to a one-dimensional inequality, which Jodeit extended to q > 1. Carleson and Chang proved that this inequality has extremals when q ≥ 2 is an integer. Henc...

We prove novel (local) square function/Carleson measure estimates for non-negative solutions to the evolutionary $ p $-Laplace equation in complement of parabolic Ahlfors-David regular sets. In case heat equation, Laplace as well corresponding function have proven fundamental symmetry and inverse/free boundary type problems, particular study (parabolic) uniform rectifiability. Though implicatio...

A simple model of Laplacian growth is considered, in which the growth takes place only at the tips of long, thin fingers. Following Carleson and Makarov [L. Carleson and N. Makarov, J. Anal. Math. 87, 103 (2002)], the evolution of the fingers is studied with use of the deterministic Loewner equation. The method is then extended to study the growth in a linear channel with reflecting sidewalls. ...

We characterize, using the Bergman kernel, Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in C, generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measures, and we compute the speed of escape to the boundary of uniformly d...

Let 1 ≤ p < ∞ and let μ be a positive finite Borel measure on the unit disk D. The area Nevanlinna-Lebesgue space Np(μ) consists of all measurable functions h on D such that log |h| ∈ Lp(μ), and the area Nevanlinna space N α is the subspace of N p((1 − |z|2)αdν(z)), where α > −1 and ν is area measure on D, consisting of all holomorphic functions. We characterize Carleson measures for N α, defin...

Hankel operators on the Hardy space of the disk, H (D) , can be studied as linear operators from H (D) to its dual space, as conjugate linear operators from H (D) to itself, or, in the viewpoint we will take here, as bilinear functionals on H (D) × H (D) . In that formulation, given a holomorphic symbol function b we consider the bilinear Hankel form, defined initially for f, g in P (D) , the s...

We prove L estimates (Theorem 1.3) for the Bi-Carleson operator defined below. The methods used are essentially based on the treatment of the Walsh analogue of the operator in the prequel [11] of this paper, but with additional technicalities due to the fact that in the Fourier model one cannot obtain perfect localization in both space and frequency.

and Applied Analysis 3 The dilations on H are defined by δz δ z′, t δz′, δ2t , δ > 0, and the rotation on H n is defined by σz σ z′, t σz′, t with a unitary map σ of C. The conjugation of z is z z′, t z′,−t . The norm function is given by |z| ∣∣z′∣∣4 |t| )1/4 , z ( z′, t ) ∈ H, 1.5 which is homogeneous of degree 1 and satisfies |z−1| |z| and |zw| ≤ C |z| |w| for some absolute constant C. The di...