Related papers: F. Wiener's trick and an extremal problem for $H^p…

This paper provides a study of problems related to Hardy spaces left by G.\,Weiss in \cite{We}. First, We will prove that the Hardy spaces $H^p(\mathbb{R}^n)$ can be characterized by a fixed Lipschitz function.

In a previous work (arXiv:2306.05948), we constructed by convex integration examples of energy dissipating solutions to the 2D Euler equations on $\mathbb{R}^2$ with vorticity in the real Hardy space $H^p(\mathbb{R}^2)$. In the present…

Two problems are posed that involve the star-invariant subspace $K^p_\theta$ (in the Hardy space $H^p$) associated with an inner function $\theta$. One of these asks for a characterization of the extreme points of the unit ball in…

We study estimates for Hardy space norms of analytic projections. We first find a sufficient condition for the Bergman projection of a function in the unit disc to belong to the Hardy space $H^p$ for $1 < p < \infty$. We apply the result to…

We analyze Bergman spaces A p f (D) of generalized analytic functions of solutions to the Vekua equation $\partial$w = ($\partial$f /f)w in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions f and 1…

This paper aims to obtain decompositions of higher dimensional $L^p(\mathbb{R}^n)$ functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range $0<p<1$. In the one-dimensional…

We show topological genericity for the set of functions in the space X, where X denotes the intersection of the Hardy spaces H^p with p<1, on the open unit disc such that the sequence of Taylor coefficients of the function and of all…

We prove the nontrivial variant \[ \sum\limits_{m,n=1}^{\infty}\Big(\frac{n}{m}\Big)^{\frac{1}{q}-\frac{1}{p}}\frac{a_mb_n}{m+n-1}\leq\frac{\pi}{\sin\frac{\pi}{p}} \Big( \sum\limits_{m=1}^{\infty}a_m^p\Big)^{\frac 1p}\Big(…

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces $H^p(\mathbf R^{n+1}_+), 1\leq p\leq \infty.$ Namely, for $f\in L^p(\mathbf R^n)$, Clifford algebra-valued, $f$ is further…

Let $\mathcal{S}_H^0(K)$, $K\ge 1$, be the class of normalized $K$-quasiconformal harmonic mappings in the unit disk. We obtain Baernstein type extremal results for the analytic and co-analytic parts of functions in the geometric subclasses…

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-H\"older continuous condition. In this article, the authors first introduce the variable weak Hardy space on $\mathbb R^n$,…

A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $\Cn$ for $0<p<\infty$ is given. When $0<p\leq1$, the result is new even in the case of the unit disk.

A Hilbert point in $H^p(\mathbb{T}^d)$, for $d\geq1$ and $1\leq p \leq \infty$, is a nontrivial function $\varphi$ in $H^p(\mathbb{T}^d)$ such that $\| \varphi \|_{H^p(\mathbb{T}^d)} \leq \|\varphi + f\|_{H^p(\mathbb{T}^d)}$ whenever $f$ is…

Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ is an absolute continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L^p(\mathbb{T})$, where $\dot{F}(e^{it}) = \frac{d}{dt} F(e^{it})$ and $p \in…

Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces $H^p$ of the upper half-plane and we review how their Fredholm properties can be studied in terms…

In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some sharp inequalities on analytic functions in the unit disk. We answer a question that…

We study the norm of point evaluation at the origin in the Paley--Wiener space $PW^p$ for $0 < p < \infty$, i. e., we search for the smallest positive constant $C$, called $\mathscr{C}_p$, such that the inequality $|f(0)|^p \leq C…

In this paper, we study an extremal problem concerning best approximation in the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szeg\"o kernel and its derivative, parametrized…

Our starting point is a theorem of de Leeuw and Rudin that describes the extreme points of the unit ball in the Hardy space $H^1$. We extend this result to subspaces of $H^1$ formed by functions with smaller spectra. More precisely, given a…

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…