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Artur Nicolau

Publications and source records attributed to Artur Nicolau.

At least 37 records · Page 2Linked to original sources
arXiv2019

Oscillation of Functions in the Hölder class

We study the size of the set of points where the $α$-divided difference of a function in the Hölder class $Λ_α$ is bounded below by a fixed positive constant. Our results are obtained from their discrete analogues which can be stated in the language of dyadic martingales. Our main technical result in this setting is a sharp estimate of the Hausdorff measure of the set of points where a dyadic martingale with bounded increments has maximal growth.

math.CA↗
arXiv2019

Invertibility Threshold for Nevanlinna Quotient Algebras

Let $\mathcal{N}$ be the Nevanlinna class and let $B$ be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra $\mathcal{N} / B \mathcal{N}$, that is, $|f| \ge e^{-H} $ on the set $B^{-1}\{0\}$ for some positive harmonic function $H$, holds if and only if the function $- \log |B|$ has a harmonic majorant on the set $\{z\in\mathbb{D}:ρ(z,Λ)\geq e^{-H(z)}\}$; at least for large enough functions $H$. We also study the corresponding class of positive harmonic functions $H$ in the unit disc such that the latter condition holds. We also discuss the analogous invertibility problem in quotients of the Smirnov class.

math.CA↗
arXiv2018

Distortion and Distribution of Sets under Inner Functions

It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions fixing the origin. In this setting, the distortion of Hausdorff contents has also been studied. We present here similar results focusing on inner functions with fixed points on the unit circle. In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. As an application, we use them to estimate the size of irregular points of inner functions omitting large sets. Finally, we also present a natural interpretation of the results in the upper half plane.

math.CV↗
arXiv2018

Approximation in the Zygmund Class

We study the distance in the Zygmund class $Λ_{\ast}$ to the subspace $\operatorname{I}(\operatorname{BMO})$ of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of $\operatorname{I}(\operatorname{BMO})$ in the Zygmund seminorm. We also generalise this result to Zygmund measures on $\mathbb{R}^d.$ Finally, we apply the techniques developed in the article to characterise the closure of the subspace of functions in $Λ_{\ast}$ that are also in the classical Sobolev space $W^{1,p},$ for $1 < p < \infty.$

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arXiv2018

The Corona Property in Nevanlinna quotient algebras and Interpolating sequences

Let $I$ be an inner function in the unit disk $\mathbb D$ and let $\mathcal N$ denote the Nevanlinna class. We prove that under natural assumptions, Bezout equations in the quotient algebra $\mathcal N/I\mathcal N$ can be solved if and only if the zeros of $I$ form a finite union of Nevanlinna interpolating sequences. This is in contrast with the situation in the algebra of bounded analytic functions, where being a finite union of interpolating sequences is a sufficient but not necessary condition. An analogous result in the Smirnov class is proved as well as several equivalent descriptions of Blaschke products whose zeros form a finite union of interpolating sequences in the Nevanlinna class.

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arXiv2018

Accessible Parts of Boundary for Simply Connected Domains

For a bounded simply connected domain $Ω\subset\mathbb{R}^2$, any point $z\inΩ$ and any $0<α<1$, we give a lower bound for the $α$-dimensional Hausdorff content of the set of points in the boundary of $Ω$ which can be joined to $z$ by a John curve with a suitable John constant depending only on $α$, in terms of the distance of $z$ to $\partialΩ$. In fact this set in the boundary contains the intersection $\partialΩ_z\cap\partialΩ$ of the boundary of a John sub-domain $Ω_z$ of $Ω$, centered at $z$, with the boundary of $Ω$. This may be understood as a quantitative version of a result of Makarov. This estimate is then applied to obtain the pointwise version of a weighted Hardy inequality.

math.CA↗
arXiv2017

On square functions with independent increments and Sobolev spaces on the line

We prove a characterization of some $L^p$-Sobolev spaces involving the quadratic symmetrization of the Calderón commutator kernel, which is related to a square function with differences of difference quotients. An endpoint weak type estimate is established for functions in homogeneous Hardy-Sobolev spaces $\dot H^1_α$. We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class.

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arXiv2016

Oscillation of generalized differences of Hölder and Zygmund functions

In this paper we analyze the oscillation of functions having derivatives in the Hölder or Zygmund class in terms of generalized differences and prove that its growth is governed by a version of the classical Kolmogorov's Law of the Iterated Logarithm. A better behavior is obtained for functions in the Lipschitz class via an interesting connection with Calderón-Zygmund operators.

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arXiv2016

Inner Functions in certain Hardy-Sobolev Spaces

For $1/2<p<1$, a description of inner functions whose derivative is in the Hardy space $H^p$ is given in terms of either their mapping properties or the geometric distribution of their zeros.

math.CV↗
arXiv2016

Finitely generated ideals in the Nevanlinna class

In this paper we investigate finitely generated ideals in the Nevanlinna class. We prove analogues to some known results for the algebra of bounded analytic functions $H^{\infty}$. We also show that, in contrast to the $H^{\infty}$-case, the stable rank of the Nevanlinna class is strictly bigger than 1.

math.CV↗
arXiv2015

Wepable Inner Functions

Following Gorkin, Mortini, and Nikolski, we say that an inner function $I$ in $H^\infty$ of the unit disc has the WEP property if its modulus at a point $z$ is bounded from below by a function of the distance from $z$ to the zero set of $I$. This is equivalent to a number of properties, and we establish some consequences of this for $H^\infty/IH^\infty$. The bulk of the paper is devoted to "wepable" functions, i.e. those inner functions which can be made WEP after multiplication by a suitable Blaschke product. We prove that a closed subset $E$ of the unit circle is of finite entropy (i.e. is a Beurling-Carleson set) if and only if any singular measure supported on $E$ gives rise to a wepable singular inner function. As a corollary, we see that singular measures which spread their mass too evenly cannot give rise to wepable singular inner functions. Furthermore, we prove that the stronger property of porosity of $E$ is equivalent to a stronger form of wepability ("easy wepability") for the singular inner functions with support in $E$. Finally, we find out the critical decay rate of masses of atomic measures (with no restrictions on support) guaranteeing that the corresponding singular inner functions are easily wepable.

math.CV↗
arXiv2015

Uniform separation through intermediate points

It is shown that a separated sequence of points in the unit disc of the complex plane is in fact uniformly separated, if there exists a certain intermediate sequence whose separated subsequences are uniformly separated. This property is applied to improve a recent result in the theory of differential equations.

math.CA↗
arXiv2014

Mean growth and geometric zero distribution of solutions of linear differential equations

The aim of this paper is to consider certain conditions on the coefficient $A$ of the differential equation $f"+Af=0$ in the unit disc, which place all normal solutions $f$ to the union of Hardy spaces or result in the zero-sequence of each non-trivial solution to be uniformly separated. The conditions on the coefficient are given in terms of Carleson measures.

math.CA↗
arXiv2014

Extremal solutions of Nevalinna-Pick problems and certain classes of inner functions

Consider a scaled Nevanlinna-Pick interpolation problem and let $Π$ be the Blaschke product whose zeros are the nodes of the problem. It is proved that if $Π$ belongs to a certain class of inner functions, then the extremal solutions of the problem or most of them, are in the same class. Three different classical classes are considered: inner functions whose derivative is in a certain Hardy space, exponential Blaschke products and also the well known class of $α$-Blaschke products, for $0<α<1$.

math.CV↗
arXiv2014

Divided Differences, Square Functions and a Law of the Iterated Logarithm

We study differentiability properties of functions defined in the euclidean space in terms of a conical square function which is analogue to the classical square function introduced by Stein and Zygmund in the sixties. Pointwise differentiability can be characterized, modulo sets of Lebesgue measure zero, in terms of the finiteness of the conical square function. At the complement of this set, a Law of the Iterated Logarithm describes the maximal growth of the divided differences in terms of a truncated version of the conical square function. Moreover a characterization of Sobolev spaces in terms of the conical square function is given.

math.CA↗
arXiv2013

Reverse Carleson measures in Hardy spaces

We give a necessary and sufficient condition for a measure $μ$ in the closed unit disk to be a reverse Carleson measure for Hardy spaces. This extends a previous result of Lefévre, Li, Queffélec and Rodr\'ıguez-Piazza \cite{LLQR}. We also provide a simple example showing that the analogue for the Paley-Wiener space does not hold. This example can be generalised to model spaces associated to one-component inner functions.

math.CV↗