Related papers: $H^1$ and dyadic $H^1$

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 prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calder\'{o}n-Zygmund operators as several functions can now…

We study boundedness properties of a class of multiparameter paraproducts on the dual space of the dyadic Hardy space H_d^1(T^N), the dyadic product BMO space BMO_d(T^N). For this, we introduce a notion of logarithmic mean oscillation on…

This paper provides a deeper study of the Hardy and $\rm BMO$ spaces associated to the Neumann Laplacian $\Delta_N$. For the Hardy space $H^1_{\Delta_N}(\mathbb{R}^n)$ (which is a proper subspace of the classical Hardy space…

Let $\theta$ be an inner function satisfying the connected level set condition of B. Cohn, and let $K^{1}_{\theta}$ be the shift-coinvariant subspace of the Hardy space $H^1$ generated by $\theta$. We describe the dual space to…

In this paper we establish a T1 criterion for the boundedness of Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space naturally associated to the Hermite operator H. We apply this criterion in a systematic way to prove the boundedness…

This note contains two simple observations. First, by the weak factorization of product $H^1$ (Ferguson--Lacey, Lacey--Terwilleger), we obtain a multi-parameter analogue of Hardy's inequality. Second, as a dual statement, the Fourier…

Multi-norm singular integrals and Fourier multipliers were introduced in [29], and one application of these notions was a precise description of the composition of convolution operators with Calder\'on-Zygmund kernels adapted to $n$…

In this paper we extend dyadic shifts and the dyadic representation theorem to an operator-valued setting: We first define operator-valued dyadic shifts and prove that they are bounded. We then extend the dyadic representation theorem,…

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 $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$…

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space $\B.$ If we denote by $H$ the Hilbert space…

Let $(h_I)$ denote the standard Haar system on $[0,1]$, indexed by $I\in \mathcal D$, the set of dyadic intervals and $h_I\otimes h_J$ denote the tensor product $(s,t)\mapsto h_I(s) h_J(t)$, $I,J\in \mathcal D$. We consider a class of…

We obtain the explicit upper Bellman function for the natural dyadic maximal operator acting from ${\rm BMO}(\mathbb{R}^n)$ into ${\rm BLO}(\mathbb{R}^n).$ As a consequence, we show that the ${\rm BMO}\to{\rm BLO}$ norm of the natural…

The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals…

We will introduce the basics of dyadic harmonic analysis and how it can be used to obtain weighted estimates for classical Calder\'on-Zygmund singular integral operators and their commutators. Harmonic analysts have used dyadic models for…

Consider the second order divergence form elliptic operator $L$ with complex bounded coefficients. In general, the operators related to it (such as Riesz transform or square function) lie beyond the scope of the Calder\'{o}n-Zygmund theory.…

Let $\Gamma$ be a graph with the doubling property for the volume of balls and $P$ a reversible random walk on $\Gamma$. We introduce $H^1$ Hardy spaces of functions and $1$-forms adapted to $P$ and prove various characterizations of these…

We prove that the $\cal P$ norm estimate between a Hardy martingale and its cosine part are stable under dyadic perturbations, and show how dyadic stability of the $\cal P$ norm estimate is used in the proof that $L^1$ embeds into…

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 Hardy space…

The paper considers the Hilbert space $\hat{H}_r$ of real functions summable with the square $L^2(a,b)_r$ on any interval $\{(a,b)_r\}_{r=1}^{\infty}\in \mathbb{R}$. It is shown on the basis of the theorem on zeros of real orthogonal…