经典分析与常微分方程

We generalize to arbitrary dimensions an example originally introduced by Besicovitch, obtaining for every $d \geq 1$ a purely $d$-unrectifiable set $E \subset \mathbb{R}^{d+1}$ such that $\Theta_{\ast}^{d}(E, x) = \liminf_{r \to 0}…

We establish the first results on $\mathrm{L}^p$ bounds for Riesz transforms associated with non-autonomous second order parabolic differential operators in divergence form with bounded coefficients that depend measurably on all variables.…

We prove a quantitative Roth theorem in the plane for the two-dimensional polynomial pattern $(x_1,x_2), (x_1,x_2)+(t_1,t_2), (x_1,x_2)+(t_1^2+t_2^2,t_1^3+t_2^3)$. A pointwise convergence result for the associated polynomial ergodic average…

We obtain an infinite-series representation for the arc length of a supercircle in terms of the scale parameter $a$ and the shape parameter $n$. The resulting expression is constructed by means of generalized binomial coefficients and Gauss…

Weighted estimates for the fractional integral operator $I_\alpha$ are established and subsequently applied to derive corresponding Hardy--Sobolev inequalities. The weights are constructed from distance functions to bounded median porous…

A set $E\subset\R$ is measure universal if every set of positive Lebesgue measure contains an affine copy of $E$. By a theorem of Bourgain, a sum of three infinite sets is never measure universal, while the two-set regime is one of the…

In this paper, we study integral operators \begin{equation*} T_\alpha f(x)=\int_{\mathbb{R}^{n}}K(x,y) f(y)dy, \end{equation*} with kernels $K(x,y)= k_1( x- A_1y)...k_m( x-A_my),$ where $k_i(x)=\frac{\Omega_i(x)}{|x|^{n/q_i}}$ and…

We prove a P\'{o}lya-Szeg\H{o} principle for the Riesz $(p,\alpha)$-variation, a scale of fractional smoothness interpolating between bounded $p$-variation and the Sobolev space $W^{1,p}$. In contrast to the classical P\'{o}lya-Szeg\H{o}…

This work concerns a quantitative form of Landau's eigenvalue theorem for spatio-spectral limiting operators. We isolate a simple mechanism that converts the problem of estimating the distribution of eigenvalues of a limiting operator into…

We prove two strict total-positivity results by isolating two strictification mechanisms. The first is a spectral Darboux mechanism: an induction converts positivity and ordered endpoint asymptotics for a one-dimensional spectral family…

We prove a Bernstein--Ganzburg type limit relation \[ \lim_{n\to\infty} \Bigl(\frac{n}{\sigma}\Bigr)^{(2a+1)/p}E_{n,\sigma}(f)_{p,a,b} =A_{\sigma}(f)_{p,a}, \] where $E_{n,\sigma}(f)_{p,a,b}$ is the error of best approximation of…

This paper develops a probabilistic sign rule for quotients of functions represented by positive series or integrals. For a function in this class, normalising the summand function in the series case or the integrand function in the…

We establish a new regularity phenomenon of continuous functions. Specifically, given any continuous function $f$ and arbitrary $\epsilon>0$, we construct a Lipschitz perturbation $g_\epsilon$ whose Lipschitz seminorm is less than…

We prove new $L^p$ boundedness results for Bochner-Riesz means associated with the spectral decomposition of the sub-Laplacian on the Heisenberg group $\mathbb H_n$. Our results hold for a range $1\le p\le p_n$ where $p_n\to 2$ as…

For an arbitrary reduced root system, we give upper bounds for the Dunkl kernel with regular spectral parameter and its derivatives, which are uniform in the spatial variable. These estimates generalize well-known sharp upper bounds for…

This paper investigates the relationship between tiles and weak tiles in the context of finite cyclic group $\mathbb{Z}_{pq}$. We prove that weak tiles and translational tiles are equivalent in this group. Our proof employs Fourier…

We study weighted estimates for linear and multilinear integral operators whose kernels satisfy only size conditions. Extending a theorem of E. Stein and its refinement by Soria and Weiss, we prove weighted estimates on Herz and Ces\`aro…

We prove a new polynomial upper bound for the Lebesgue constants of $\tau$-Leja sequences on finite unions of real intervals. Building on an estimate of Andrievskii and Nazarov, we replace the global separation of the first $n$ Leja points…

In this paper we study the summability of solutions of some general forms of singularly perturbed linear ordinary differential and moment differential equations. We conclude that under some assumptions solutions of these equations are…

We refine the method by Greenfeld and Lev for the product spectral set problem and generalize the theorem to a singular measure setting. Furthermore, we establish a new class of spectral unions of intervals for which the product spectral…