经典分析与常微分方程

In this paper, we investigate the $H^p(G) \rightarrow L^p(G)$, $0< p \leq 1$, boundedness of multiplier operators defined via group Fourier transform on a graded Lie group $G$, where $H^p(G)$ is the Hardy space on $G$. Our main result…

The main content of this text is composed from the two doctoral dissertations of V.\,V.~Katrakhov (1989) and S.\,M.~Sitnik (2016). Apart from a detailed review and bibliography the book contains original results of the authors. The text is…

We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…

Let n be an odd integer greater than 1. We slice a circular pizza into 2n slices, making cuts from a non-central interior point of the circle. We estimate the difference between between the total area of the even numbered slices and the…

We present a systematic approach to the problem whether a topologically infinite-dimensional space can be made homogeneous in the Coifman--Weiss sense. The answer to the examined question is negative, as expected. Our leading representative…

Let $\gamma:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(\gamma(\theta),\gamma'(\theta),\gamma''(\theta)\big)\neq 0$. For each $\theta\in[0,1]$, let…

In this paper we study the restriction estimate for the flat disk over finite fields. Mockenhaupt and Tao initially studied this problem but their results were addressed only for dimensions $n=4,6$. We improve and extend their results to…

The operator that intertwines between the $\mathbb{Z}_2$ - Dunkl operator and the derivative is shown to have a realization in terms of the oscillator operators in one dimension. This observation rests on the fact that the Dunkl…

Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections'' of a "network" corresponding to a fractal set, $F$. This lead to the definition of the…

Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has…

The aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way, and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials we associate…

We generalize a result of McDonald and Taylor which concerns the size of the tuples of edge lengths in the set $C_1 \times C_2$ utilizing the notion of thickness. Specifically, we show that $C_1, C_2 \subset \mathbb{R}^d$ compact sets with…

We study the problem of whether the centered Hardy--Littlewood maximal function of a singular function is absolutely continuous. For a parameter $d \in (0,1)$ and a closed set $E\subset [0,1]$, let $\mu$ be a $d$-Ahlfors regular measure…

Let $\mathbf{x}=(x_1,\dots,x_n)$ be an $n$-tuple of positive real numbers and the sequence $(x_i)_{i\in\mathbb{Z}}$ be its $n$-periodic extension. Given an $n$-tuple $\mathbf{r}=(r_1,\dots,r_n)$ of positive integers, let $a_i$ be the…

Let $\Omega$ be a convex open set in $\mathbb R^n$ and let $\Lambda_k$ be a finite subset of $\Omega$. We find necessary geometric conditions for $\Lambda_k$ to be interpolating for the space of multivariate polynomials of degree at most…

We show that the problem of finding the measure supported on a compact subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to K is a minimum, is…

We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential…

Kurdyka-{\L}ojasiewicz (K{\L}) functions are real-valued functions characterized by a differential inequality involving the norm of their gradient. This class of functions is quite rich, containing objects as diverse as subanalytic,…

We consider a generalized version of the sign uncertainty principle for the Fourier transform, first proposed by Bourgain, Clozel and Kahane in 2010 and revisited by Cohn and Gon\c{c}alves in 2019. In our setup, the signs of a function and…

If $f(x,y)$ is a real function satisfying $y>0$ and $\sum_{r=0}^{n-1}f(x+ry,ny)=f(x,y)$ for $n=1,2,3,\ldots$, we say that $f(x,y)$ is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz…