经典分析与常微分方程

We present the construction of a complete system of functions associated with the Sturm-Liouville equation in impedance form on a finite interval $I$, given an impedance function $a\in L^2(I)$. The system, known as the formal powers, is…

This paper investigates the existence of positive solutions for regular discrete second-order single-variable boundary value problems with mixed boundary conditions, including a nonhomogeneous Dirichlet boundary condition, of the form:…

An alternative method to invert the Radon transforms without the use of Courant-Hilbert's identities has been proposed and developed independently from the space dimension. For the universal representation of inverse Radon transform, we…

Using real-variable methods, we characterise multipliers for general classes of Hardy--Orlicz spaces, unifying and extending several classical results due to Hardy and Littlewood; Duren and Shields; Paley; and others. Applications of our…

Let f be an integrable function which has integral 0 on R n. What is the largest condition on |f | that guarantees that f is in the Hardy space H 1 (R n)? When f is compactly supported, it is well-known that it is necessary and sufficient…

We provide quantitative weighted weak type estimates for non-integral square functions in the critical case $p=2$ in terms of the $A_p$ and reverse H\"older constants associated to the weight. The method of proof uses a decoupling of the…

A new Goodman-Sharma type modification of the Meyer-K\"{o}nig and Zeller operator for approximation of bounded continuous functions on [0,1) is presented. We estimate the approximation error of the proposed operator and prove direct and…

In this paper, we study the almost everywhere convergence of sequences of two-parameter ergodic averages over rectangles in the plane. On the one hand, we show that if the rectangles we consider have their sides with slopes in a finitely…

In this paper, we study the stability of the concentration inequality for one-dimensional complex polynomials. We provide the stability of the local concentration inequality and a global version using a Wehrl-type entropy.

We prove a fractal uncertainty principle with exponent $\frac{d}{2} - \delta + \varepsilon$, $\varepsilon > 0$, for Ahlfors--David regular subsets of $\mathbb R^d$ with dimension $\delta$ which satisfy a suitable "nonorthogonality…

Given any (forward) Brascamp--Lieb inequality on euclidean space, a famous theorem of Lieb guarantees that gaussian near-maximizers always exist. Recently, Barthe and Wolff used mass transportation techniques to establish a counterpart to…

Let \(\mu\) be a finite Borel measure on \((-\pi,\pi)\). Consider the one-dimensional Poisson equation \(-u''=\mu\), where equality holds in the sense of distributions, with Dirichlet boundary conditions \(u(\pm\pi)=0\). In this paper, we…

In our recent work [SIGMA \textbf{20} (2024), 074, 13 pages], the leading behaviour of the Humbert function $\Psi_1[a,b;c,c';x,y]$ when $x\to\infty$ and $y\to +\infty$ has been derived in a direct and simple manner. In this paper, we obtain…

We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to $0^+$. We show that dyadic expansions are…

We construct solutions of the $2$-dimensional Toda-Hirota equation (2dTHE) expressed by the Gelfand hypergeometric function (Gelfand HGF) on the Grassmannian $\mathrm{GM}(2,N)$ of confluent or non-confluent type, which is labeled by a…

We present an astonishingly simple and elegant proof of the celebrated Basel problem.

Let $H\subset \R^{d+1}$ be a compact, convex, analytic hypersurface of finite type with a smooth measure $\sigma $ on $H$. Let $\kappa$ denote the Gaussian curvature on $H$. We consider the oscillatory integral $(\kappa^{1/2}…

We precisely characterize the relationships between the reverse H\"older inequality, the Fujii-Wilson condition, the B\'ekoll\'e-Bonami $\mathrm{B}_p$ condition, the $\mathrm{B}_\infty$ condition, and the reverse Jensen inequality, for…

In this paper, we study sequences of positive numbers preserving summability. In particular, the open set property for such a family of sequences is shown. Several classes of sequences preserving summability, including polynomials, sums of…

We prove that the classical Laplace asymptotic expansion (AE) of $\int_{\mathbb R^d} g(x)e^{-nu(x)}dx$, $n\gg1$ extends to the high-dimensional regime in which $d$ may grow large with $n$. More specifically, we use new techniques suitable…