Related papers: Note on potential theory for functions in Hardy cl…

We develop the compactness theory of multilinear singular integrals on product spaces using a modern point of view. The first main result is a compact $T1$ theorem for multilinear Calder\'{o}n--Zygmund operators on product spaces. More…

We characterize the space of multipliers from the Hardy space of Dirichlet series $\mathcal H_p$ into $\mathcal H_q$ for every $1 \leq p,q \leq \infty$. For a fixed Dirichlet series, we also investigate some structural properties of its…

Three density theorems for three suitable subspaces of $SBD$ functions, in the strong $BD$ topology, are proven. The spaces are $SBD$, $SBD^p_\infty$, where the absolutely continuous part of the symmetric gradient is in $L^p$, with $p>1$,…

We consider several problems at or beyond endpoint in harmonic analysis. The solutions of these problems are related to the estimates of some classes of sublinear operators. To do this, we introduce some new functions spaces…

In this paper we find the condition on function $\omega$ and weight $v$ which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces ${\mathcal…

In this paper we prove off-diagonal, limited range, multilinear, vector-valued, and two-weight versions of the Rubio de Francia extrapolation theorem in general quasi-Banach function spaces. We prove mapping properties of the generalization…

We study properties of ridge functions $f(x)=g(a\cdot x)$ in high dimensions $d$ from the viewpoint of approximation theory. The considered function classes consist of ridge functions such that the profile $g$ is a member of a univariate…

In this note besides two abstract versions of the Vitali Covering Lemma an abstract Hardy-Littlewood Maximal Inequality, generalizing weak type (1,1) maximal function inequality, associated to any outer measure and a family of subsets on a…

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series $D(\alpha;z)=\sum_{n\geq 2}(\log n)^{\alpha}(\eta_n+{\rm i} \theta_n)/n^z$, properly scaled and normalized, where…

In this paper, we provide an efficient method for computing the Taylor coefficients of $1-p_n f$, where $p_n$ denotes the optimal polynomial approximant of degree $n$ to $1/f$ in a Hilbert space $H^2_\omega$ of analytic functions over the…

Let $n\ge 1$ and $\varphi: \mathbb{D}^n\to\mathbb{D}$ be a holomorphic function, where $\mathbb{D}$ denotes the open unit disk of $\mathbb{C}$. Let $\Theta: \mathbb{D} \to \mathbb{D}$ be an inner function and $K^p_\Theta$, $p>0$, denote the…

Let $X$ be a metric space equipped with a doubling measure. We consider weights $w(x)=\operatorname{dist}(x,E)^{-\alpha}$, where $E$ is a closed set in $X$ and $\alpha\in\mathbb R$. We establish sharp conditions, based on the Assouad…

Let $X = \left\{x_1, \dots, x_N\right\} \subset \mathbb{T}^d \cong [0,1]^d$ be a set of $N$ points in the $d-$dimensional torus that we want to arrange as regularly possible. The purpose of this paper is to introduce a curious energy…

The direct and inverse theorems are established for the best approximation in the weighted $L^p$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups. The theorems are stated…

We study non-analytic terms, which cannot be written in the form of any positive integer power of field-dependent mass squared, in effective potential at finite temperature in one-loop approximation for a real scalar field on the…

We show, by applying discrete weighted norm inequalities and the Rubio de Francia algorithm, that the discrete Hilbert transform and discrete Riesz potential are bounded on variable $\ell^{p(\cdot)}(\mathbb{Z})$ spaces whenever the discrete…

We prove sharp version of Riesz-Fej\'er inequality for functions in harmonic Hardy space $h^p(\mathbb{D})$ on the unit disk $\mathbb{D}$, for $p>1,$ thus extending the result from \cite{KPK} and resolving the posed conjecture.

The celebrated Hardy inequality can be written in the form $$\int_0^\infty \mathcal{P}_p \big(f|_{[0,x]}\big)dx \le (1-p)^{-1/p} \int_0^\infty f(x)\:dx \qquad \text{ for }p\in(0,1)\text{ and }f \in L^1\text{ with }f\ge0,$$ where…

We extend certain classical theorems in pluripotential theory to a class of functions defined on the support of a $(1,1)$-closed positive current $T$, analogous to plurisubharmonic functions, called $T$-plurisubharmonic functions. These…

We show that the norm in the Hardy space $H^p$ satisfies \begin{equation}\label{absteq} \|f\|_{H^p}^p\asymp\int_0^1M_q^p(r,f')(1-r)^{p\left(1-\frac1q\right)}\,dr+|f(0)|^p\tag{\dag} \end{equation} for all univalent functions provided that…