Artur Nicolau

BMO for nondoubling measures

1. Introduction. The Calderón-Zygmund theory of singular integrals has been traditionally considered with respect to a measure satisfying a doubling condition. Recently, Tolsa [T] and, independently, Nazarov, Treil, and Volberg [NTV] have shown that this standard doubling condition was not really necessary. Likewise, in the homogeneous spaces setting, functions of bounded mean oscillation, BMO, and its predual H 1 , the atomic Hardy space, play an important role in the theory of singular integrals. This note is an attempt to find good substitutes for the spaces BMO and H 1 when the underlying measure is nondoubling. Our hope was that we would have been able to prove some results of Tolsa, Nazarov, Treil, and Volberg, via BMO-H 1 interpolation, but in this respect we were unsuccessful. Let µ be a nonnegative Radon measure on R n. A function f ∈ L 1 loc (µ) is said to belong to BMO(µ) if the inequality

Inner Functions, Bloch Spaces and Symmetric Measures

Schwarzs lemma asserts that analytic mappings from the unit disc into itself decrease hyperbolic distances. In this paper, inner functions which decrease hyperbolic distances as much as possible, when one approaches the unit circle, are constructed. Actually, it is shown that a quadratic condition governs the best decay of the hyperbolic derivative of an inner function. This is related to a result of L. Carleson on the existence of singular symmetric measures. As a consequence, some results on composition operators are obtained, bringing out the importance of the Bloch spaces in this connection. Another consequence is a uniform way of producing singular measures which are simultaneously symmetric and Kahane. 1991 Mathematics Subject Classification: primary 30D50; secondary 30D45, 26A30, 47B38.

Frostman shifts of inner functions

In this paper we study the classM of all inner functions whose non-zero Frostman shifts are Carleson-Newman Blaschke products. We present several geometric, measure theoretic and analytic characterizations ofM in terms of level sets, distribution of zeros, and behaviour of pseudohyperbolic derivatives and observe thatM is the set of all functions inH∞ whose range on the set of trivial points in the maximal ideal space is ∂D∪{0}

The corona property for bounded analytic functions in some Besov spaces

In this paper, the corona theorem for the algebra of bounded analytic functions in the unit disc which are in the Besov space Bp, 1 ( > 0, z E A there exist g1, .. E Ho n BPA such that (2) fig + 1 . ..... 1fngn... Received by the editors May 22, 1989. 1980 Mathematics Suibject Classification (1985 Revision). Primary 30A98; Secondary 46J15, 46J20.

A note on interpolation in the Hardy spaces of the unit disc

In this note we formulate and solve a natural interpolation problem for the Hardy spaces in the unit disc in terms of maximal functions and weighted summable sequences.

Regularity Properties of Measures, Entropy and the Law of the Iterated Logarithm

We study regularity properties of a positive measure in euclidean space, such as being absolutely continuous with respect to certain Hausdorff measures, in terms of their dyadic doubling properties. Applications of the main results to the distortion of homeomorphisms of the real line and to the regularity of the harmonic measure for some degenerate elliptic operators are given.

The closure of the Hardy space in the Bloch norm

A description of the closure in the Bloch norm of the Bloch functions that are in a Hardy space is given. The result uses the classical estimates for the Lusin area function. §

Interpolating Blaschke products solving Pick-Nevanlinna problems

We study when a Pick-Nevanlinna problem with more than one solution can be solved by an interpolating Blaschke product.

Differentiability of functions in the Zygmund class

Let f be a function in the Zygmund class in the euclidean space. It is proved that the Hausdorff dimension of the set of points where f has bounded divided differences, is bigger or equal to one. Furthermore, if f is in the Small Zygmund class, then the Hausdorff dimension of the set of points where f is differentiable, is bigger or equal to one. The sharpness of these results is also discussed.

Multiplicative Square Functions

We study regularity properties of a positive measure in the euclidean space in terms of two square functions which are the multiplicative analogues of the usual martingale square function and of the Lusin area function of a harmonic function. The size of ...