经典分析与常微分方程
A first-order ordinary differential equation, solved with respect to derivative, is considered. It's right-hand side is defined and continuous on the set, consisting of a connected open subset of a two-dimensional Euclidean space and a part…
We introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We then use this notion to define the quaternionic Fredholm determinant of trace-class operators in…
We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin.…
This paper deals with a BMO Theorem for $\epsilon$ distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$ with applications to manifolds of speech and sound. The material for this paper appears in the research memoir [2].
In this announcement we consider the following problem. Let $n,m\geq 1$, $U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $\phi:U\to \mathbb R^n$ be a…
The "visibility" of a planar set $S$ from a point $a$ is defined as the normalized size of the radial projection of $S$ from $a$ to the unit circle centered at $a$. Simon and Solomyak (Real Anal. Exchange 2006/07) proved that unrectifiable…
We show that Nikodym sets and local smoothing estimates for linear wave equations form a dichotomy: If Nikodym sets for a family of curves exist, then the related maximal operator is not bounded on $L^p(\mathbb{R}^2)$ for any $p<\infty$; if…
We give different integral representations of the Lommel function $s_{\mu,\nu}(z)$ involving trigonometric and hypergeometric $_2F_1$ functions. By using classical results of Polya, we give the distribution of the zeros of $s_{\mu,\nu}(z)$…
This article serves as a study guide for the $\ell^2$ decoupling theorem for the paraboloid originally proved by Bourgain and Demeter. Given its popularity and importance, many expositions about the $\ell^2$ decoupling theorem already…
This article deals with maximal operators on ${\mathbb R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional H\"ormander--Mihlin multiplier with the identity in $n-d$ coordinates, in the particular codimension 1…
In this paper, we studied the space-time estimates for the solution to the Schr\"odinger equation. By polynomial partitioning, induction arguments, bilinear to linear arguments and broad norm estimates, we set up several maximal estimates…
The long-standing conjecture that for $p \in (1, \infty)$ the $\ell^p(\mathbb Z)$ norm of the Riesz--Titchmarsh discrete Hilbert transform is the same as the $L^p(\mathbb R)$ norm of the classical Hilbert transform, is verified when $p = 2…
The purpose of this article is to describe the singularities of one-dimensional oscillatory integrals, whose phases have a certain singularity, in the form of an asymptotic expansion. In the case of the Laplace integral, an analogous result…
We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function $d(n)$, that is, for a measure preserving $\mathbb Z[i]$ action $T$, the limit $$\lim_{N\rightarrow \infty} \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n)…
In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum…
This expository piece expounds on major themes and clarifies technical details of the paper "Kaufman and Falconer estimates for radial projections and a continuum version of Beck's theorem" of Orponen, Shmerkin, and Wang.
The paper contains a survey of the results obtained by the author in recent years. These results concern the application in multivariate polynomial interpolation of some geometric constructions and methods. In particular, we give estimates…
The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this…
In this article we aim to obtain the Fisher Riemann geodesics for nonparametric families of probability densities as a weak limit of the parametric case with increasing number of parameters.
Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace…