Related papers: The Real Characterization of $H_{\lambda}^p(\mathb…

We obtain sharp two-sided inequalities between $L^p-$norms $(1<p<\infty)$ of functions $Hf$ and $H^*f$, where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty).$ In an equivalent form,…

In this paper, we show the equivalence between the boundedness of the Riesz transform $d\Delta^{-1/2}$ on $L^p$, $p\in (2,p_0)$, and the equality $H^p=L^p$, $p\in(2,p_0)$, in the class of manifold whose measure is doubling and for which the…

Let $d \in \{3, 4, 5, \ldots\}$ and $p \in (0,1]$. We consider the Hermite operator $L = -\Delta + |x|^2$ on its maximal domain in $L^2(\mathbb{R}^d)$. Let $H_L^p(\mathbb{R}^d)$ be the completion of $ \{ f \in L^2(\mathbb{R}^d):…

In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on a Euclidean space $\mathbb{R}^N$. On the half-space $\mathbb{R}_+\times\mathbb{R}^N$,…

We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of…

Let $L$ be a one-to-one operator of type $\omega$ in $L^2(\mathbb{R}^n)$, with $\omega\in[0,\,\pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$…

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Let $L=-\Delta +V$, $V\geq 0$, be the Dunkl--Schr\"odinger operator on $\mathbb R^N$. Assume that there…

Let $0<p<\infty$, $\Gamma$ be a Lipschitz curve on the complex plane~$\mathbb{C}$ and $\Omega_+$ is the domain above $\Gamma$, we define Hardy space $H^p(\Omega_+)$ as the set of analytic functions $F$ satisfying…

Let $\Delta$ be the Dunkl Laplacian on $\mathbb R^N$ associated with a normalized root system $R$ and a multiplicity function $k(\alpha)\geq 0$. We say that a function $f$ belongs to the Hardy space $H^1_{\Delta}$ if the nontangential…

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 $\Gamma$ be a Lipschitz curve on the complex plane $\mathbb{C}$ and $\Omega_+$ is the domain above $\Gamma$, we define Hardy space $H^p(\Omega_+)$ as the set of holomorphic functions $F$ satisfying $\sup_{\tau>0}(\int_{\Gamma}…

Let $L=-\Delta+V$ be a Schr\"odinger operator acting on $L^2(\mathbb R^n)$, $n\ge1$, where $V\not\equiv 0$ is a nonnegative locally integrable function on $\mathbb R^n$. In this article, we will introduce weighted Hardy spaces $H^p_L(w)$…

In this work we study the vector-valued Hardy spaces H p (D; F) (1 \leq p \leq \infty) and their relationship with RNP, ARNP and the UMDP properties. By following the approach of Taylor in the scalar-valued case, we prove that, when F and F…

On $\mathbb{R}^N$ equipped with a normalized root system $\mathcal R$ and a multiplicity function $k\geq 0$, let $dw(\mathbf x)=\Pi_{\alpha\in \mathcal R}|\langle \mathbf x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x$,…

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…

Let $M$ be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces $H^p$ of differential forms on $M$ and give various characterizations of them, including an atomic decomposition.…

A characterization is obtained for those pairs of weights $v$ and $w$ on $\mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(\mathbb{R}^2_+)$ to…

Let $\mathcal{L}_k = -\Delta_k + V$ be a Schr\"odinger operator associated with the Dunkl Laplacian $\Delta_k$, where $V$ is the non-negative potential function belonging to the reverse H\"older class $RH_k^q(\mathbb{R}^n)$ with $q>…

The main result of this paper are dimension-free $L^p$ inequalities, $1<p<\infty$, for low degree scalar-valued functions on the Hamming cube. More precisely, for any $p>2,$ $\varepsilon>0,$ and $\theta=\theta(\varepsilon,p)\in (0,1)$…

This is the second in our series of papers concerning some reversed Hardy--Littlewood--Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb…