Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt [12]. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an application we discuss how distributional estimates are obtained for the linear and bilinear Hilbert transform. Th...

Let A be a uniform algebra on the compact space X and σ a probability measure on X . We define the Hardy spaces H(σ) and the H(σ) interpolating sequences S in the p-spectrum Mp of σ. We prove, under some structural hypotheses on σ that ”Carleson type” conditions on S imply that S is interpolating with a linear extension operator in H(σ), s < p provided that either p = ∞ or p ≤ 2. This gives new...

The purposes of this paper are two fold. First, we extend the method of non-homogeneous harmonic analysis of Nazarov, Treil and Volberg to handle “Bergman–type” singular integral operators. The canonical example of such an operator is the Beurling transform on the unit disc. Second, we use the methods developed in this paper to settle the important open question about characterizing the Carleso...

The purposes of this paper are two fold. First, we extend the method of non-homogeneous harmonic analysis of Nazarov, Treil and Volberg to handle “Bergman–type” singular integral operators. The canonical example of such an operator is the Beurling transform on the unit disc. Second, we use the methods developed in this paper to settle the important open question about characterizing the Carleso...

We consider the quadratic family of maps given by fa(x) = 1− ax with x ∈ [−1, 1], where a is a Benedicks-Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes X0, X1, . . ., given by Xn = f n a , for every integer n ≥ 0, where each random variable Xn is distributed according to the unique absolutely continu...

In this work we prove Hardy space estimates for bilinear Littlewood-Paley-Stein square function and Calderón-Zygmund operators. Sufficient Carleson measure type conditions are given for square functions to be bounded from H p1 ×H p2 into Lp for indices smaller than 1, and sufficient BMO type conditions are given for a bilinear Calderón-Zygmund operator to be bounded from H p1 ×H p2 into H p for...

Let H(b) denote the de Branges–Rovnyak space associated with a function b in the unit ball of H∞(C+). We study the boundary behavior of the derivatives of functions in H(b) and obtain weighted norm estimates of the form ‖f ‖L2(μ) ≤ C‖f‖H(b), where f ∈ H(b) and μ is a Carleson-type measure on C+ ∪ R. We provide several applications of these inequalities. We apply them to obtain embedding theorem...