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We obtain several new results on the simultaneous packing and covering constant $\gamma(\mathcal{X})$ of a Banach space $\mathcal{X}$, and its lattice counterpart $\gamma^*(\mathcal{X})$. These constants measure how efficient a (lattice)…

It is well-known that a random variable, i.e., a function defined on a probability space, with values in a Borel space, can be represented on the special probability space consisting of the unit interval with Lebesgue measure. We show an…

In his recent research, the author improved the error term in the prime geodesic theorem for compact, even-dimensional, rank one locally symmetric spaces. It turned out that the obtained estimate $O(x^{2\rho-\frac{\rho}{n}}(\log x)^{-1})$…

We study an infinite dimensional analysis with respect to the measure on Schwartz space of tempered distributions, corresponding to the distributional derivative of gamma process. Laguerre polynomials being orthogonal with respect to gamma…

In this note, we study a quantitative extension of the John-Nirenberg inequality for the Hardy-Littlewood maximal function of a $\operatorname{BMO}$ function. More precisely, for every nonconstant locally integrable function $f$ such that…

The research problem in this work is the relaxation of maximizing non-negative submodular plus modular with the entire real number domain as its value range over a family of down-closed sets. We seek a feasible point $\mathbf{x}^*$ in the…

Let ${\mathcal X}$ be a space of homogeneous type in the sense of Coifman and Weiss and ${\mathcal D}$ a collection of balls in $\cx$. The authors introduce the localized atomic Hardy space $H^{p, q}_{\mathcal D}({\mathcal X})$ with $p\in…

Bounded Oscillation (BO) operators were recently introduced in the author's paper [13], where it was proved that many operators in harmonic analysis (Calder\'on-Zygmund operators, Carleson type operators, martingale transforms,…

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class…

In this paper we consider a random entire function of the form $f(z,\omega )=\sum\nolimits_{n=0}^{+\infty}\xi_n(\omega )a_nz^n,$ where $\xi_n(\omega )$ are independent standard\break complex gaussian random variables and $a_n\in\mathbb{C}$…

Using notions from the geometry of Banach spaces we introduce square functions $\gamma(\Omega,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the…

The generalized Boole transformations have rich behavior ranging from the \textit{mixing} phase with the Cauchy invariant measure to the \textit{dissipative} phase through the \textit{infinite ergodic} phase with the Lebesgue measure. In…

In this paper we prove maximal regularity estimates in "square function spaces" which are commonly used in harmonic analysis, spectral theory, and stochastic analysis. In particular, they lead to a new class of maximal regularity results…

A scale of the Frechet spaces of exponential type entire functions of one complex variable is considered. Certain special properties of subsets of these spaces consisting of Laguerre entire functions, which are obtained as uniform limits on…

In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace-Beltrami operator on the unit sphere. The sufficient conditions for summability is obtained. The more general…

The purpose of the paper is to study the operators on the weighted Bergman spaces on the unit disk ${\mathbb{D}}$, denoted by $A^{p}_{\lambda,w}({\mathbb{D}})$, that are associated with a class of generalized analytic functions, named the…

We propose a simple uniform lower bound on the spacings between the successive zeros of the Laguerre polynomials $L_n^{(\alpha)}$ for all $\alpha>-1$. Our bound is sharp regarding the order of dependency on $n$ and $\alpha$ in various…

Let $0 \leq \alpha<n$ and $b$ be the locally integrable function. In this paper, we consider the maximal commutator of fractional maximal function $M_{b,\alpha}$ and the nonlinear commutator of fractional maximal function $[b, M_{\alpha}]$…

We introduce Bourgain-Morrey-Lorentz spaces and give a description of the predual of Bourgain-Morrey-Lorentz spaces via the block spaces. As an application of duality, we obtain the boundedness of Hardy-Littlewood maximal operator, sharp…

Let $(\mathcal{X}, d, \mu)$ be a metric measure space, $L$ a linear operator which has a bounded $H_\infty$ functional calculus and satisfies the Davies-Gaffney estimate, $\Phi$ a concave function on $(0,\infty)$ of critical lower type…