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Artur Nicolau

Publications and source records attributed to Artur Nicolau.

47 records · Page 3Linked to original sources
arXiv2013

Oscillation of Hölder Continuous Functions

Local oscillation of a function satisfying a Hölder condition is considered and it is proved that its growth is governed by a version of the Law of the Iterated Logarithm.

math.CA↗
arXiv2012

Inner Functions with Derivatives in the Weak Hardy Space

It is proved that exponential Blaschke products are the inner functions whose derivative is in the weak Hardy space. Exponential Blaschke products are described in terms of their logarithmic means and also in terms of the behavior of the derivatives of functions in the corresponding model space.

math.CA↗
arXiv2010

Paths of inner-related functions

We characterize the connected components of the subset $\cni$ of $H^\infty$ formed by the products $bh$, where $b$ is Carleson-Newman Blaschke product and $h\in H^\infty$ is an invertible function. We use this result to show that, except for finite Blaschke products, no inner function in the little Bloch space is in the closure of one of these components. Our main result says that every inner function can be connected with an element of $\cni$ within the set of products $uh$, where $u$ is inner and $h$ is invertible. We also study some of these issues in the context of Douglas algebras.

math.CA↗
arXiv2010

Differentiability of functions in the Zygmund class

We prove that the Hausdorff dimension of the set of points where a function in the Zygmund class in the euclidean space has bounded divided differences, is bigger or equal to 1. A similar result for functions in the Small Zygmund class is also proved and an example is constructed to show that these results are sharp.

math.CA↗
arXiv2010

Smoothness of sets in Euclidean spaces

We study some properties of smooth sets in the sense defined by Hungerford. We prove a sharp form of Hungerford's Theorem on the Hausdorff dimension of their boundaries on Euclidean spaces and show the invariance of the definition under a class of automorphisms of the ambient space.

math.CA↗
arXiv2003

Harmonic and superharmonic majorants on the disk

We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain logarithmic Lipschitz upper envelope of it (relevant because of the Harnack inequality) admits a superharmonic majorant. We discuss the logarithmic Lipschitz regularity of this superharmonic majorant, and show that in general it cannot be better than that of the Poisson kernel. We provide examples to show that mere superharmonicity of the data does not help with the problem of existence of a harmonic majorant.

math.CV↗
arXiv2003

The constant of interpolation

We prove that a suitably adjusted version of Peter Jones' formula for interpolation by bounded holomorphic functions gives a sharp upper bound for what is known as the constant of interpolation. We show how this leads to precise and computable numerical bounds for this constant.

math.CV↗
arXiv2002

Interpolating sequences for the Nevanlinna and Smirnov classes

We give an analytic characterisation of the interpolating sequences for the Nevanlinna and Smirnov classes. From this we deduce a necessary and a sufficient geometric condition, both expressed in terms of a certain non-tangential maximal function associated to the sequence. Some examples show that the gap between the necessary and the sufficient condition cannot be covered. We also discuss the relationship between our results and the previous work of Naftalevic for the Nevanlinna class, and Yanagihara for the Smirnov class. Finally, we observe that the arguments used in the previous proofs show that interpolating sequences for ``big'' Hardy-Orlicz spaces are in general different from those for the scale included in the classical Hardy spaces.

math.CV↗