经典分析与常微分方程

This paper investigates the restricted circular planar three-body problem. We prove that for every negative Jacobi constant of sufficiently large magnitude, the surface of unperturbed parabolic solutions breaks to induce homoclinic tangle…

We obtain several degenerations of the $q$-Heun equation by considering the linear $q$-difference equations associated to several $q$-Painlev\'e equations. We establish definitions of the confluent $q$-Heun equation, the biconfluent…

The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group $G$ and a finite subset $F$ of $G$, that if there is a set $A$ that solves the tiling equation $\mathbb{1}_F * \mathbb{1}_A = 1$, there is also a periodic…

The quantum annulus of type $r$ is the class of invertible operators with singular values in $(1/r,r).$ Given an analytic function on the classical annulus of type $r,$ we may evaluate it on operators in the quantum annulus by The spectral…

The main goal of this note is to characterize the necessary and sufficient conditions for a composition operator to act between spaces of mappings of bounded Wiener variation in a normed-valued setting. The necessary and sufficient…

In this paper, we study the $T$-periodic solutions of the parameter-dependent $\phi$-Laplacian equation \begin{equation*} (\phi(x'))'=F(\lambda,t,x,x'). \end{equation*} Based on the topological degree theory, we present some atypical…

The operators $\Lambda_m$ ($m\in\mathbb{N}\cup \{0\}$) arise when one studies the action of the Beurling-Ahlfors transform on certain radial function subspaces. It is known that the weak-type $(1,1)$ constant of $\Lambda_0$ is equal to…

We discuss the closability of directional derivative operators with respect to a general Radon measure $\mu$ on $\mathbb{R}^d$; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from…

We derive double-product representations of nonterminating basic hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We refer to this result as the $q$-Chaundy theorem and several limiting $q\to…

In this survey we summarize the current state of known orthogonality relations for the $q$ and $q^{-1}$-symmetric and dual subfamilies of the Askey--Wilson polynomials in the $q$-Askey scheme. These polynomials are the continuous dual $q$…

We study the problem of differentiation of integrals for certain bases in the infinite-dimensional torus $\mathbb{T}^\omega$. In particular, for every $p_0 \in [1,\infty)$, we construct a basis $\mathcal{B}$ which differentiates…

Let $\mu_1$ and $\mu_2$ be two complex-valued Borel measures on the real line such that $\operatorname{supp} \mu_1 =[\alpha_1,\beta_1] < \operatorname{supp} \mu_2 =[\alpha_2,\beta_2]$ and ${\rm d}\mu_i(x) = -\rho_i(x){\rm d}x/2\pi {\rm i}$,…

We construct a compact set in $\mathbb R^2$ of measure 0 containing a piece of a parabola of every aperture between 1 and 2. As a consequence, we improve lower bounds for the $L^p$-$L^q$ norm of the corresponding maximal operator for a…

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,…

This work presents two simple criteria for determining the oscillatory nature of solutions to second-order differential equations with deviated arguments. These criteria extend the (Leighton-Wintner)-type criteria established by G.Q. Wang…

Exact solutions of both the Navier-Stokes and Euler equations are found on the surface of a sphere. Under the assumption of a vanishing convection term, the flow of two oppositely rotating point vortices at the poles turns out to be the…

We consider the asymptotic evaluation of the integral transform $\int_0^\infty f(x) \, \sin^n(\lambda x)/x^n \,\text{d} x$ of an exponential type function $f(x)$ of type $\tau>0$, for large values of the parameter $\lambda$, where $n$ is a…

Using a high/low argument, we prove a universal $\ell^2L^6$ decoupling estimate with constant $C_\epsilon R^{\epsilon}$ for general convex curves in the plane. These curves have no additional regularity assumptions, and the constant…

This paper studies the structure of Kakeya sets in $\mathbb{R}^3$. We show that for every Kakeya set $K\subset\mathbb{R}^3$, there exist well-separated scales $0<\delta<\rho\leq 1$ so that the $\delta$ neighborhood of $K$ is almost as large…

For the $\alpha$-Green kernel $g^\alpha_D$ on a domain $D\subset\mathbb R^n$, $n\geqslant2$, associated with the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, where $\alpha\in(0,n)$ and $\alpha\leqslant2$, and a relatively closed set $F\subset…