arXiv subjects

Valentina Ciccone

Publications and source records attributed to Valentina Ciccone.

12 recordsLinked to original sources

arXiv2026

On high dimensional maximal functions associated to Gaussians, balls, and spheres

We prove that for each $p\in (1,\infty),$ the norms on $L^p(\mathbb{R}^d)$ of the maximal functions associated to Gaussians (heat semigroup), balls (Hardy-Littlewood averages), and spheres (spherical averages) converge, as the dimension $d\to \infty,$ to the same quantity $λ(p)$. This is derived from the fact that the norms on $L^2(\mathbb{R}^d)$ of the maximal functions corresponding to the differences of Gaussian, ball, and spherical averages converge to zero with the dimension $d.$ The fact is proved with the aid of estimates for Fourier multiplier symbols corresponding to these averages, a general principle that allows us to control the norm of a maximal function corresponding to a Fourier multiplier operator by the norm of the multiplier operator itself, and concentration properties of high dimensional Gaussian random vectors. Moreover, relying on the properties of the $d$-dimensional maximal function for the heat semigroup $\mathcal{G}_\ast^d$, we show that $λ(p)$ satisfies $$ \frac25\frac{p}{p-1}\le\|\mathcal{G}_\ast^1\|_{L^p(\mathbb{R})\rightarrow L^p(\mathbb{R})}\le λ(p)\le \frac{p}{p-1}. $$ In particular, to obtain the middle inequality we show that the norms on $L^p(\mathbb{R}^d)$ of the maximal function for the heat semigroup are non-decreasing in $d.$

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arXiv2026

High-dimensional limits and extremizers for maximal functions associated with log-concave densities

We introduce a unified framework to establish the high-dimensional asymptotic behavior of maximal functions associated with radial log-concave probability densities, encompassing the maximal heat semigroup, Hardy-Littlewood maximal function over Euclidean balls, and, additionally, maximal spherical means. Namely, for any $p \in (1, \infty)$, we prove that the $L^p(\mathbb{R}^d)$ operator norms of these maximal operators all converge as the dimension $d \to \infty$ to a single, universal limit $λ(p)$. Furthermore, by proving that the $L^p$ operator norms for the heat semigroup $\mathcal G_*^d$ are monotonically non-decreasing in the dimension, we provide explicit quantitative bounds on the universal limit, showing that $\frac{2}{5}\frac{p}{p-1} \le \|\mathcal{G}_*^1\|_{L^p(\mathbb{R}) \to L^p(\mathbb{R})} \le λ(p) \le \frac{p}{p-1}$. We also prove an extremality property: among all symmetric convex bodies in high dimensions, the maximal operator associated with the Euclidean ball achieves the asymptotically minimal $L^p$ operator norm. Our main results are established via a general transference principle that allows us to control maximal functions via Fourier multiplier symbols. To estimate these symbols uniformly across log-concave densities, we import variance type bounds and thin-shell type concentration of measure results, which are novel tools in the study of maximal functions. In particular, to prove the extremality property, we require a variance type bound for general log concave measures established in a recent series of breakthroughs in high dimensional convex geometry.

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arXiv2025

Global and local maximizers for some Fourier extension estimates on the sphere

In this note, we study maximizers for Fourier extension inequalities on the sphere. We prove that constant functions are local maximizers for the $L^p(\mathbb{S}^{d-1})$ to $L^p(\mathbb{R}^d)$ Fourier extension estimates in the same range of exponents $p$ for which they are global maximizers for the $L^2(\mathbb{S}^{d-1})$ to $L^p_{rad}L^2_{ang}(\mathbb{R}^d)$ mixed-norm Fourier extension inequalities. Moreover, in the case of low dimensions, we improve the range of exponents for which constant functions are known to be the unique global maximizers for the $L^2(\mathbb{S}^{d-1})$ to $L^p_{rad}L^2_{ang}(\mathbb{R}^d)$ mixed-norm Fourier extension estimate on the sphere, covering, for the case of dimensions $d=2,3$, the entire Stein-Tomas range. This is achieved by establishing novel hierarchies between certain weighted norms of Bessel functions.

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arXiv2025

A note on bilinear multipliers with convex singularities

We study bounds in the local $L^2$ range of exponents for bilinear multipliers whose symbol is the characteristic function of the epigraph of certain convex curves. We realize these bounds as a consequence of estimates that we establish, via simple arguments, for the associated exotic paraproducts. As a further application, we observe bounds beyond the local $L^2$ range for bilinear multipliers whose symbol is the characteristic function of the epigraph of convex polygonal curves associated with these paraproducts.

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arXiv2024

Endpoint estimates for higher order Marcinkiewicz multipliers

We consider Marcinkiewicz multipliers of any lacunary order defined by means of uniformly bounded variation on each lacunary Littlewood--Paley interval of some fixed order $τ\geq 1$. We prove the optimal endpoint bounds for such multipliers as a corollary of a more general endpoint estimate for a class of multipliers introduced by Coifman, Rubio de Francia and Semmes and further studied by Tao and Wright. Our methods also yield the best possible endpoint mapping property for higher order Hörmander-Mihlin multipliers, namely multipliers which are singular on every point of a lacunary set of order $τ$. These results can be considered as endpoint versions of corresponding results of Sjögren and Sjölin. Finally our methods generalize a weak square function characterization of the space $L\log^{1/2}L$ in terms of a square function introduced by Tao and Wright: we realize such a weak characterization as the dual of the Chang--Wilson--Wolff inequality, thus giving corresponding weak square function characterizations for the spaces $L\log^{τ/2}L$ for general integer orders $τ\geq 1$.

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arXiv2024

Singular multipliers on multiscale Zygmund sets

Given an Orlicz space $ L^2 \subseteq X \subseteq L^1$ on $[0,1]$, with submultiplicative Young function ${\mathrm{Y}_X}$, we fully characterize the closed null sets $Ξ$ of the real line with the property that Hörmander-Mihlin or Marcinkiewicz multiplier operators $\mathrm{T}_m$ with singularities on $Ξ$ obey weak-type endpoint modular bounds on $X$ of the type \[ \left|\left\{x\in \mathbb R : |\mathrm{T}_m f(x)| >λ\right\}\right| \leq C \int_{\mathbb R} \mathrm{Y}_X \left(\frac{|f|}λ\right), \qquad \forall λ>0. \] These sets $Ξ$ are exactly those enjoying a scale invariant version of Zygmund's $(L\sqrt{\log L},{L^2})$ improving inequality with $X$ in place of the former space, which is termed multiscale Zygmund property. Our methods actually yield sparse and quantitative weighted estimates for the Fourier multipliers $\mathrm{T}_m$ and for the corresponding square functions. In particular, our framework covers the case of singular sets $Ξ$ of finite lacunary order and thus leads to modular and quantitative weighted versions of the classical endpoint theorems of Tao and Wright for Marcinkiewicz multipliers. Moreover, we obtain a pointwise sparse bound for the Marcinkiewicz square function answering a recent conjecture of Lerner. On the other hand, examples of non-lacunary sets enjoying the multiscale Zygmund property for each $X=L^p$, $1<p\leq 2$ are also covered. The main new ingredient in the proofs is a multi-frequency, multi-scale projection lemma based on Gabor expansion, and possessing independent interest.

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arXiv2024

Sharp Fourier Extension on the Circle Under Arithmetic Constraints

We establish a sharp adjoint Fourier restriction inequality for the end-point Tomas-Stein restriction theorem on the circle under a certain arithmetic constraint on the support set of the Fourier coefficients of the given function. Such arithmetic constraint is a generalization of a $B_3$-set.

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arXiv2020

Regularized transport between singular covariance matrices

We consider the problem of steering a linear stochastic system between two end-point degenerate Gaussian distributions in finite time. This accounts for those situations in which some but not all of the state entries are uncertain at the initial, t = 0, and final time, t = T . This problem entails non-trivial technical challenges as the singularity of terminal state-covariance causes the control to grow unbounded at the final time T. Consequently, the entropic interpolation (Schroedinger Bridge) is provided by a diffusion process which is not finite-energy, thereby placing this case outside of most of the current theory. In this paper, we show that a feasible interpolation can be derived as a limiting case of earlier results for non-degenerate cases, and that it can be expressed in closed form. Moreover, we show that such interpolation belongs to the same reciprocal class of the uncontrolled evolution. By doing so we also highlight a time-symmetry of the problem, contrasting dual formulations in the forward and reverse time-directions, where in each the control grows unbounded as time approaches the end-point (in the forward and reverse time-direction, respectively).

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arXiv2019

Robust Identification of "Sparse Plus Low-rank" Graphical Models: An Optimization Approach

Motivated by graphical models, we consider the "Sparse Plus Low-rank" decomposition of a positive definite concentration matrix -- the inverse of the covariance matrix. This is a classical problem for which a rich theory and numerical algorithms have been developed. It appears, however, that the results rapidly degrade when, as it happens in practice, the covariance matrix must be estimated from the observed data and is therefore affected by a certain degree of uncertainty. We discuss this problem and propose an alternative optimization approach that appears to be suitable to deal with robustness issues in the "Sparse Plus Low-rank" decomposition problem.The variational analysis of this optimization problem is carried over and discussed.

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arXiv2018

Factor Models with Real Data: a Robust Estimation of the Number of Factors

Factor models are a very efficient way to describe high dimensional vectors of data in terms of a small number of common relevant factors. This problem, which is of fundamental importance in many disciplines, is usually reformulated in mathematical terms as follows. We are given the covariance matrix Sigma of the available data. Sigma must be additively decomposed as the sum of two positive semidefinite matrices D and L: D | that accounts for the idiosyncratic noise affecting the knowledge of each component of the available vector of data | must be diagonal and L must have the smallest possible rank in order to describe the available data in terms of the smallest possible number of independent factors. In practice, however, the matrix Sigma is never known and therefore it must be estimated from the data so that only an approximation of Sigma is actually available. This paper discusses the issues that arise from this uncertainty and provides a strategy to deal with the problem of robustly estimating the number of factors.

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arXiv2018

An alternating minimization algorithm for Factor Analysis

The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem. This algorithm appears to perform extremely well and is extremely fast even when the given covariance matrix has a very large dimension. The effectiveness of the algorithm is assessed through simulation studies and by applications to three real datasets that are considered as benchmark for the problem. A local convergence analysis of the algorithm is also presented.

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arXiv2017

Factor analysis with finite data

Factor analysis aims to describe high dimensional random vectors by means of a small number of unknown common factors. In mathematical terms, it is required to decompose the covariance matrix $Σ$ of the random vector as the sum of a diagonal matrix $D$ | accounting for the idiosyncratic noise in the data | and a low rank matrix $R$ | accounting for the variance of the common factors | in such a way that the rank of $R$ is as small as possible so that the number of common factors is minimal. In practice, however, the matrix $Σ$ is unknown and must be replaced by its estimate, i.e. the sample covariance, which comes from a finite amount of data. This paper provides a strategy to account for the uncertainty in the estimation of $Σ$ in the factor analysis problem.

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