Related papers: Regularity properties of the $\alpha$-Wilton funct…

A plasmon of a bounded domain $\Omega\subset\mathbb{R}^n$ is a non-trivial bounded harmonic function on $\mathbb{R}^n\setminus\partial\Omega$ which is continuous at $\partial\Omega$ and whose exterior and interior normal derivatives at…

We consider a model of Branching Brownian Motion with time-inhomogeneous variance of the form \sigma(t/T), where \sigma is a strictly decreasing function. Fang and Zeitouni (2012) showed that the maximal particle's position M_T is such that…

For $0<\alpha<1$ let $V(\alpha)$ denote the supremum of the numbers $v$ such that every $\alpha$-H\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2…

In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main…

We study the regularity of Radon measures $\mu$ which satisfy that there exists a function $h_\mu$ in $H^1(\Omega)$, stationary harmonic such that $\Delta h_\mu =\mu$ in $\Omega$ (here $\Omega$ is an open set of $\mathbb{R}^2$). Such…

Let $(\mathbb{X} , d, \mu )$ be a proper metric measure space and let $\Omega \subset \mathbb{X}$ be a bounded domain. For each $x\in \Omega$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial \Omega ) $ and let $B_x$ be the…

The metrical theory of the product of consecutive partial quotients is associated with the uniform Diophantine approximation, specifically to the improvements to Dirichlet's theorem. Achieving some variant forms of metrical theory in…

This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators, $$\mathfrak{M}_{\alpha,\Omega}(\vec{f})(x)=\sup\limits_{0<r<{\rm…

Minkowski's question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in…

Solutions to a wide variety of transcendental equations can be expressed in terms of the Lambert $\mathrm{W}$ function. The $\mathrm{W}$ function, occurring frequently in applications, is a non-elementary, but now standard mathematical…

We continue the study of the fractional variation following the distributional approach developed in the previous works arXiv:1809.08575, arXiv:1910.13419 and arXiv:2011.03928. We provide a general analysis of the distributional space…

Fix an integer $s$. Let $f:[0,1)^s \to \mathbb R$ be an integrable function. Let $P\subset [0,1]^s$ be a finite point set. Quasi-Monte Carlo integration of $f$ by $P$ is the average value of $f$ over $P$ that approximates the integration of…

The cross-Wigner distribution $W(f,g)$ of two functions or temperate distributions $f,g$ is a fundamental tool in quantum mechanics and in signal analysis. Usually, in applications in time-frequency analysis $f$ and $g$ belong to some…

We study the Sobolev regularity on the sphere $\mathbb{S}^d$ of the uncentered fractional Hardy-Littlewood maximal operator $\widetilde{\mathcal{M}}_{\beta}$ at the endpoint $p=1$, when acting on polar data. We first prove that if…

We study the covariance property of quadratic time-frequency distributions with respect to the action of the extended symplectic group. We show how covariance is related, and in fact in competition, with the possibility of damping the…

We study the additive functional $X_n(\alpha)$ on conditioned Galton-Watson trees given, for arbitrary complex $\alpha$, by summing the $\alpha$th power of all subtree sizes. Allowing complex $\alpha$ is advantageous, even for the study of…

For a possibly singular subset of a regular Poisson manifold we construct a deformation quantization of its algebra of Whitney functions. We then extend the construction of a deformation quantization to the case where the underlying set is…

Let $q\ge 2$ and $N\ge 1$ be integers. W. Zhang (2008) has shown that for any fixed $\epsilon> 0$, and $q^{\epsilon} \le N \le q^{1/2 -\epsilon}$, $$ \sum_{\chi \ne \chi_0} |\sum_{n=1}^N \chi(n)|^2 |L(1, \chi)|^2 = (1 + o(1)) \alpha_q q N…

We study commutators of the Riesz potential $I_\alpha$ with functions $b$ in the capacitary space $\mathrm{BMO}^\beta(\mathbb{R}^n)$, defined through the Hausdorff content $\mathcal{H}^\beta_\infty$. We prove a Chanillo-type theorem…

Let $ 0\le f\in C^{0,1}(\mathbb R^n)$. Given a domain $\Omega\subset \mathbb R^n$, we prove that any stable solution to the equation $-\Delta u=f(u)$ in $\Omega$ satisfies a BMO interior regularity when $n=10$, and an Morrey…