arXiv subjects

Nicola Arcozzi

Publications and source records attributed to Nicola Arcozzi.

At least 19 recordsLinked to original sources

arXiv2026

Dyadic potential theory and de Rham functions

We study de Rham functional equations driven by two increasing fractional linear transformations. Our main purpose is to relate the singularity theory of the associated solutions to dyadic potential theory on the binary tree. We first prove an existence and uniqueness theorem for increasing, left-continuous solutions in the full range of linear fractional data, and identify the trapping region in parameter space where the solution is continuous. For a large class of parameters we show that the de Rham solution is the normalized cumulative capacitary function of a multiplicative dyadic capacity. This gives a potential-theoretic model for Möbius de Rham systems. We then sharpen Okamura's Hausdorff-dimensional estimates for the singular measure associated with the solution by replacing Hausdorff dimension with dyadic Riesz capacities at the upper endpoint of Okamura's theorem.

math.AP↗

arXiv2025

Functional Analysis and Operator Theory

Lecture note topics: 1. Some tools from real and complex analysis, 2. Hilbert spaces, 3. Banach spaces, 4. Compact operators and their spectra, 5. Intermezzo: reproducing kernel Hilbert spaces, 6. Banach algebras ,7. Spectral theory of unitary, and of self-adjoint operators

math.FA↗

arXiv2025

A simplified directional KeRF algorithm

Random forest methods belong to the class of non-parametric machine learning algorithms. They were first introduced in 2001 by Breiman and they perform with accuracy in high dimensional settings. In this article, we consider, a simplified kernel-based random forest algorithm called simplified directional KeRF (Kernel Random Forest). We establish the asymptotic equivalence between simplified directional KeRF and centered KeRF, with additional numerical experiments supporting our theoretical results.

math.ST↗

arXiv2024

Paley--Wiener theorems on the Siegel upper half-space

In this paper we study spaces of holomorphic functions on the Siegel upper half-space $\mathcal U$ and prove Paley-Wiener type theorems for such spaces. The boundary of $\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. Using the group Fourier transform on $\mathbb H_n$, Ogden-Vagi proved a Paley-Wiener theorem for the Hardy space $H^2(\mathcal U)$. We consider a scale of Hilbert spaces on $\mathcal U$ that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury-Arveson space, and the Dirichlet space $\mathcal D$. For each of these spaces, we prove a Paley-Wiener theorem, some structure theorems, and provide some applications. In particular we prove that the norm of the Dirichlet space modulo constants $\dot{\mathcal D}$ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of $\mathcal U$.

math.CV↗

arXiv2023

Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc

We characterize the Carleson measures for the Dirichlet space on the bidisc, hence also its multiplier space. Following Maz'ya and Stegenga, the characterization is given in terms of a capacitary condition. We develop the foundations of a bi-parameter potential theory on the bidisc and prove a Strong Capacitary Inequality. In order to do so, we have to overcome the obstacle that the Maximum Principle fails in the bi-parameter theory.

math.CV↗

arXiv2023

Improved convergence rates for some kernel random forest algorithms

Random forests are notable learning algorithms first introduced by Breinman in 2001, they are widely used for classification and regression tasks and their mathematical properties are under ongoing research. We consider a specific class of random forest algorithms related to kernel methods, the so-called KeRF (Kernel Random Forests.) In particular, we investigate thoroughly two explicit algorithms, designed independently of the data set, the centered KeRF and the uniform KeRF. In the present article, we provide an improvement in the rate of convergence for both algorithms and we explore the related reproducing kernel Hilbert space defined by the explicit kernel of the centered random forest.

math.ST↗

arXiv2022

Two-weight dyadic Hardy's inequalities

We present various results concerning the two-weight Hardy's inequality on infinite trees. Our main scope is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization based on a new reverse Hölder inequality for trace measures, and one based on the well known Muckenhoupt-Wheeden-Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy's inequality, we characterize the compactness of the Hardy operator, we provide a list of open problems and suggest some possible lines of future research.

math.CA↗

arXiv2021

Riesz capacities of a set due to Dobiński

We study the Riesz $(a,p)$-capacity of the so called Dobiński set. We characterize the values of the parameters $a$ and $p$ for which the $(a,p)$-Riesz capacity of the Dobiński set is positive. In particular we show that the Dobiński set has positive logarithmic capacity, thus answering a question of Dayan, Fernandéz and González. We approach the problem by considering the dyadic analogues of the Riesz $(a,p)$-capacities which seem to be better adapted to the problem.

math.CA↗

arXiv2021

The Drury--Arveson space on the Siegel upper half-space and a von Neumann type inequality

In this work we study what we call Siegel--dissipative vector of commuting operators $(A_1,\ldots, A_{d+1})$ on a Hilbert space $\mathcal H$ and we obtain a von Neumann type inequality which involves the Drury--Arveson space $DA$ on the Siegel upper half-space $\mathcal U$. The operator $A_{d+1}$ is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup $\{e^{-iτA_{d+1}}\}_{τ<0}$. We then study the operator $e^{-iτA_{d+1}}A^α$ where $A^α=A_1^{α_1}\cdots A^{α_d}_d$ for $α\in\mathbb N^d_0$ and prove that can be studied by means of model operators on a weighted $L^2$ space. To prove our results we obtain a Paley--Wiener type theorem for $DA$ and we investigate some multiplier operators on $DA$ as well.

math.FA↗

arXiv2021

Ahlfors regular spaces have regular subspaces of any dimension

We characterize $Q$-dimensional Ahlfors regular spaces among trees' boundaries and show how to construct, for each $0 < α< Q$, an $α$-regular subspace. As an application, we give an alternative simple proof of the existence of $α$-regular subspaces of a $Q$-dimensional complete Ahlfors regular metric space $(X,ρ)$, which was proved in \cite{JJKRRS}.

math.MG↗

arXiv2020

Equilibrium measures on trees

We give a characterization of equilibrium measures for $p$-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For $p=2$, this provides, in the special case of trees, a converse to a theorem of Benjamini and Schramm, which interpretes the equilibrium measure of a planar graph's boundary in terms of square tilings of cylinders.

math.CA↗

arXiv2020

The Hardy space from an engineer's perspective

We give an overview of parts of the theory of Hardy spaces from the viewpoint of signals and systems theory. There are books on this topic, which dates back to Bode, Nyquist, and Wiener, and that eventually led to the developement of $H^{\infty}$ optimal control. Our modest goal here is giving a beginner's dictionary for mathematicians and engineers who know little of either systems or $H^2$ spaces.

math.CV↗

arXiv2020

Bi-parameter Carleson embeddings with product weights

Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding for a scale of Dirichlet spaces from the bi-torus to the bi-disc is equivalent to a simple ``box'' condition, for product weights on the bi-disc and arbitrary weights on the bi-torus. This gives a new simple necessary and sufficient condition for the embedding of the whole scale of weighted Dirichlet spaces of holomorphic functions on the bi-disc. This scale of Dirichlet spaces includes the classical Dirichlet space on the bi-disc. Our result is in contrast to the classical situation on the bi-disc considered by Chang and Fefferman, when a counterexample due to Carleson shows that the ``box'' condition does not suffice for the embedding to hold. But this was the embedding of bi-harmonic functions in bi-harmonic Hardy class. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures.

math.AP↗

arXiv2019

Bi-parameter embedding and measures with restriction energy condition

Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on bi-tree. In this note we give one more proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree (in a pervious paper of the authors) that used the Bellman function technique, the proof here is based on some rather subtle comparison of energies of measures on bi-tree.

math.CA↗

arXiv2018

Bellman function sitting on a tree

In this note we give a proof-by-formula of certain important embedding inequalities on dyadic tree. This is done with the help of Bellman function. We also consider the case of a bi-tree, where a different approach is explained.

math.CA↗

arXiv2018

Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-trees

In this note we investigate the multi-parameter Potential Theory on the weighted $d$-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish some basic properties of the respective potentials, capacities and equilibrium measures (in particular in the case of product polynomial weights). We explore multi-parameter Hardy inequality and its trace measures, and discuss some open problems of potential-theoretic and combinatorial nature.

math.CV↗

arXiv2018

On a class of shift-invariant subspaces of the Drury-Arveson space

In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in the complement of a set $Y\subset\mathbb{N}^d$ with the property that $Y+e_j\subset Y$ for all $j=1,\dots,d$. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitely calculated. Moreover, every such a space can be seen as an intersection of kernels of Hankel operators, whose symbols can be explicity calcuated as well. Finally, this is the right space on which Drury's inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.

math.CV↗

arXiv2017

Discrete Hilbert Transform a la Gundy-Varopoulos

We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on stochastic representation of the Hilbert transform in the continuous setting.

math.PR↗