arXiv subjects

Anthony Gauvan

Publications and source records attributed to Anthony Gauvan.

10 recordsLinked to original sources

arXiv2025

Operator capacity, the Brascamp--Lieb inequality and geometric programming

The capacity of completely positive operators and the Brascamp--Lieb constant can both be interpreted in terms of unconstrained geometric programming up to an additional minimisation over a compact group. We shine light on this perspective and make use of it to make novel contributions in both directions. For example, by making use of recent work of Bennett--Bez--Buschenhenke--Cowling--Flock, we prove new results regarding near-minimisers and local H\"older regularity of operator capacity. In addition, we observe that these results may be extended to the more general notion of capacity of quiver data. Furthermore, the geometric programming viewpoint allows us to give a new proof of the finiteness characterisation of the Brascamp--Lieb constant due to Bennett--Carbery--Christ--Tao (assuming Lieb's theorem on gaussian saturation).

math.FA↗

arXiv2024

A note on ubiquity of geometric Brascamp-Lieb data

Relying substantially on work of Garg, Gurvits, Oliveira and Wigderson, it is shown that geometric Brascamp--Lieb data are, in a certain sense, ubiquitous. This addresses a question raised by Bennett and Tao in their recent work on the adjoint Brascamp--Lieb inequality.

math.CA↗

arXiv2023

Perron's capacity of random sets

Given a sequence of random variables $\left\{ X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions $$\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}$$ and $$\Omega_{\text{rand},\text{lac}} := \left\{ \frac{ \pi X_k}{2^k} : k\geq 1 \right\}.$$ We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$.

math.FA↗

arXiv2023

Maximal operators on hyperbolic triangles

We characterize the boundedness properties on the spaces $L^p(\mathbb{H}^2)$ of the maximal operator $M_\mathcal{B}$ where $\mathcal{B}$ is an arbitrary family of hyperbolic triangles stable by isometries.

math.CA↗

arXiv2023

Sharp weak-type estimate for maximal operators associated to Cartesian families under an arithmetic condition

Given a set of integers $A \subset \mathbb{Z}$, we consider the smallest family $\mathcal{B}_{A^{n-1}}$ invariant by translation which contains the rectangles $$ R_{\boldsymbol{a}} = I_{a_1} \times \dots \times I_{a_{n-1}} \times I_{-(a_1+\dots+a_{n-1})}$$ for any $\boldsymbol{a} = (a_1,\dots,a_{n-1}) \in A^{n-1}$ and where $I_k = [0,2^k]$ for $k$ integer. We prove that if the set $A$ contains arbitrary large arithmetic progression then the maximal operator $M_{\mathcal{B}_{A^{n-1}}}$ associated to the family $\mathcal{B}_{A^{n-1}}$ is sharply bounded from $L^1\left(1+ \log^+ L^1 \right)^{n-1}$ to $L^{1,\infty}$.

math.CA↗

arXiv2022

Almost everywhere convergence for Lebesgue differentiation processes along rectangles

In this paper, we study Lebesgue differentiation processes along rectangles $R_k$ shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in $L^p$ spaces. In particular, classes of examples of such processes failing to converge a.e. in $L^\infty$ are provided, for which $R_k$ is known to be oriented along the slope $k^{-s}$ for $s>0$, yielding an interesting counterpart to the fact that the directional maximal operator associated to the set $\{k^{-s}:k\in\mathbb{N}^*\}$ fails to be bounded in $L^p$ for any $1\leq p<\infty$.

math.CA↗

arXiv2022

(Un)boundedness of directional maximal operators through a notion of "Perron capacity'' and an application

We introduce the notion of \textit{Perron capacity} of a set of slopes $\Omega \subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of $\Omega$ is finite then the directional maximal operator associated $M_\Omega$ is not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$. This allows us to prove that the set $$\Omega_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}: n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos.

math.CA↗

arXiv2022

Application of Perron Trees to Geometric Maximal Operators

We characterize the $L^p(\mathbb{R}^2)$ boundeness of the geometric maximal operator $M_{a,b}$ associated to the basis $\mathcal{B}_{a,b}$ ($a,b > 0$) which is composed of rectangles $R$ whose eccentricity and orientation is of the form $$\left( e_R ,\omega_R \right) = \left( \frac{1}{n^a} , \frac{\pi}{4n^b} \right)$$ for some $n \in \mathbb{N}^*$. The proof involves \textit{generalized Perron trees}, as constructed in \cite{KATHRYN JAN}.

math.CA↗

arXiv2022

Restricting directions for Kakeya sets

We prove that the Kakeya maximal conjecture is equivalent to the $\Omega$-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{\'e} proved that the Kakeya conjecture is equivalent to the $\Omega$-Kakeya conjecture. Moreover, we improve concrete bound on the Hausdorff dimension of a $\Omega$-Kakeya set : for any Bore set $\Omega$ in S n--1 , we prove that if X $\subset$ R n contains for any e $\in$ $\Omega$ a unit segment oriented along e then we have dX $\ge$ 6 11 d$\Omega$ + 1 where dE denotes the Hausdorff dimension of a set E.

math.CA↗

arXiv2022

Kakeya-type sets for Geometric Maximal Operators

Given a family G of rectangles, to which one associates a tree [G], one defines a natural number $\lambda$ [G] called its analytic split and satisfying, for all 1 < p < $\infty$ log($\lambda$ [G]) p MG p p where MG is the Hardy-Littlewood type maximal operator associated to the family G. As an application, we completely characterize the boundeness of planar rarefied directional maximal operators on L p for 1 < p < $\infty$. Precisely, if $\Omega$ is an arbitrary set of angles in [0, $\pi$ 4), we prove that any rarefied basis B of the directional basis R $\Omega$ yields an operator MB that has the same L p-behavior than the directional maximal operator M $\Omega$ for 1 < p < $\infty$.

math.CA↗