Related papers: From dyadic $\Lambda_{\alpha}$ to $\Lambda_{\alpha…

Let $\psi $ be a conformal map on $\mathbb{D}$ with $ \psi \left( 0 \right)=0$ and let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$ for $\alpha >0$. Denote by ${H^p}\left( \mathbb{D}…

Let $L$ be a one-to-one operator of type $\omega$ having a bounded $H_\infty$ functional calculus and satisfying the $k$-Davies-Gaffney estimates with $k\in{\mathbb N}$. In this paper, the authors introduce the weak Hardy space…

We consider weighted $L^p$-Hardy inequalities involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result…

We prove various equivalent characterisations of the Hardy space $H^p_{\mathcal{L}}(\mathbb{C}^n)$ for $0<p<1$ associated with the twisted Laplacian $\mathcal{L}$ which generalises the result of [MPR81] for the case $p=1$. Using the atomic…

Consider the wave equation associated with the Kohn Laplacian on groups of Heisenberg type. We construct parametrices using oscillatory integral representations and use them to prove sharp $L^p$ and Hardy space regularity results.

In this paper, we first obtain the operator norms of the $n$-dimensional Hardy-Littlewood-P\'{o}lya operator $\mathcal{H}$ from weighted Lebesgue spaces $L^p( \mathbb{R} ^n,| x |^{\beta} ) $ to weighted weak Lebesgue spaces…

Let P be a linear, second order, elliptic operator satisfying a Hardy inequality with potential W (i.e. $P-W\geq0$) and best constant $\alpha$. We give conditions so that the spectrum of $W^{-1}P$ is $[\alpha,\infty)$. We apply this to…

In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy--Littlewood constants for $2$-homogeneous polynomials on $\ell_p^2$ spaces, $2<p\leq\infty$…

For $p>p_0=\frac{2\lambda}{2\lambda+1}$ with $\lambda>0$, the Hardy space $H_{\lambda}^p(\mathbb{R}_+^2)$ associated with the Dunkl transform $\mathcal{F}_\lambda$ and the Dunkl operator $D$ on the real line $\mathbb{R}$, where…

In this paper, Hardy type operator $H_{\beta}$ on $\bR^{n}$ and its adjoint operator $H_{\beta}^{*}$ are investigated. We use novel methods to obtain two main results. One is that we obtain the operators $H_{\beta}$ and $H_{\beta}^{*}$…

Let $\lambda>0$, $p\in((2\lz+1)/(2\lz+2), 1]$, and $\triangle_\lambda\equiv-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$ be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces $H^p((0,…

We consider a stochastic Cahn-Hilliard partial differential equation driven by a space-time white noise. We prove the Large Deviations Principle (LDP) for the law of the solutions in the H\"older norm. We use the weak convergence approach…

Hardy spaces in the complex plane and in higher dimensions have natural finite-dimensional subspaces formed by polynomials or by linear maps. We use the restriction of Hardy norms to such subspaces to describe the set of possible…

The classical Beurling-Helson-Lowdenslager theorem characterizes the shift-invariant subspaces of the Hardy space $H^{2}$ and of the Lebesgue space $L^{2}$. In this paper, which is self-contained, we define a very general class of norms…

We establish sharp $L^p$ integral mean estimates for $(\alpha,\beta)$-harmonic functions on the unit disk. Explicit bounds for the functions and their partial derivatives are obtained in terms of boundary data, by means of the associated…

We provide the weak factorization of the Hardy spaces $H^{p}(\mathbb{R}_+, dm_{\lambda})$ in the Bessel setting, for $p\in \left(\frac{2\lambda + 1}{2\lambda + 2}, 1\right]$. As a corollary we obtain a characterization of the boundedness of…

Let $\alpha$ be a fixed quadratic irrational. Consider the Diophantine equation \[ y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where $(q_N)_{N\,\geq\,0}$ is the sequence of convergent…

This paper describes the known results on the projection from the most general holomorphic spaces $A^p_\omega$, which depend on a functional parameter $\omega$ and are over the unit disc, upper half-plane and the finite complex plane, to…

We show that if the Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the dyadic Hilbert transform, with a linear relation of the norms.

For $p\in (1,\infty)$ and $\alpha\in\mathbb{R}$, we consider measurable functions $g$ on $\mathbb{S}^{N-1}$ that satisfy the following weighted Hardy inequality: \begin{equation}\label{abs} \int_{\mathbb{R}^N}\frac{ g…