经典分析与常微分方程

Let $\nu=(\nu_1,\ldots,\nu_n)\in (-1,\vc)^n$, $n\ge 1$, and let $\mathcal{L}_\nu$ be a self-adjoint extension of the differential operator \[ L_\nu := \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(\nu_i^2 -…

In this paper, we extend the classical Newton-Maclaurin inequalities to functions $S_{k;s}(x)=E_k(x)+\dsum_{i=1}^s \al_i E_{k-i}(x)$, which are formed by linear combinations of multiple basic symmetric mean. We proved that when the…

Given a relatively compact set $\Omega \subseteq \mathbb{R}$ of Lebesgue measure $|\Omega|$ and $\varepsilon > 0$, we show the existence of a set $\Lambda \subseteq \mathbb{R}$ of uniform density $D (\Lambda) \leq (1+\varepsilon) |\Omega|$…

This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating…

We analyze, mainly using bifurcation methods, an elliptic superlinear problem in one-dimension with periodic boundary conditions. One of the main novelties is that we follow for the first time a bifurcation approach, relying on a…

Let \(\mathcal{L}_\nu\) be the Laguerre differential operator which is the self-adjoint extension of the differential operator \[ L_\nu := \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2} \left(\nu_i^2 -…

We examine the fractional heat diffusion equations $L_{\gamma,a}:=(-\Delta_a)^{\frac{\gamma}{2}}+\partial_t$, where $\Delta_a$ is the Laplace- or the Bessel-Laplace operator. We give conditions for removability which are sufficient and…

We study a family of fractional integral operators defined in $\mathbb{R}^3$ whose kernels are distributions associated with Zygmund dilations: $(x_1, x_2, x_3) \rightarrow (\delta_1 x_1, \delta_2 x_2, \delta_1\delta_2 x_3)$ for…

We introduce an extension of the generalized Riemann scheme for Fuchsian ordinary differential equations in the case of KZ-type equations. This extension describes the local structure of equations obtained by resolving the singularities of…

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $\Omega\subset \mathbb{R}^d$. This new modulus of smoothness is…

We prove a new Bernstein type inequality in $L^p$ spaces associated with the tangential derivatives on the boundary of a general compact $C^2$-domain. We give two applications: Marcinkiewicz type inequality for discretization of $L^p$ norm…

In classical analysis, the convergence behavior of power series solutions to differential or recurrence equations is generally assumed to be invariant under internal rearrangement. This paper challenges that belief by proving that, for…

In this paper, we characterize the weighted infinitesimal boundedness: for $0<\alpha<n$ and $1<p<\infty$, $$\|V\phi\|_{L^{p}(w)}^{p}\leq\epsilon\|(-\Delta)^{\frac{\alpha}{2}}\phi\|_{L^{p}(w)}^{p}+C(\epsilon)\|\phi\|_{L^{p}(w)}^{p}.$$ In…

A generalization of the classical Sard theorem in the plane is the following. Let $f$ be a function defined on a subset $A\subset{\mathbb R}^2$. If $f$ has modulus of continuity $\omega(r)\lesssim r^2$, then $f(A)\subset{\mathbb R}$ has…

In this paper, we prove that isotropic Gaussian functions are \textit{characterized} by a rearrangement inequality for weighted perimeter in dimensions $n \geq 2$ within the class of non-negative weights in $L^1(\mathbb{R}^n) \cap…

Using topological methods, we study the structure of the set of forced oscillations of a class of parametric, implicit ordinary differential equations with a generalized $\Phi$-Laplacian type term. We work in the Carath\'eodory setting.…

This note gives a bound on the error of the leading term of the $t\to 0$ asymptotic expansion of the Hartman-Watson distribution $\theta(r,t)$ in the regime $rt=\rho$ constant. The leading order term has the form…

The conference Unexpected Phenomena in Energy Minimization and Polarization, held in Sofia, Bulgaria in 2024, provided a platform for researchers to discuss and propose challenging open questions across various fields, such as potential…

In this paper, by using recurrence method and new properties of beta function and Ramanujan $R$-function, we prove the absolute monotonicity result for a function related to Ramanujan asymptotic formula of zero-balanced hypergeometric…

This expository article on the Lagrange singular integral contains two novelties. The first novelty involves a connection between the Lagrange singular integral for a simplified Clairaut equation, and Euler's homogeneous function theorem.…