Related papers: Factorization in Haar system Hardy spaces

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…

We give a constructive proof of the factorization theorem for the classical Hardy space in terms of fractional integral operator. Moreover, the result is extended to the multilinear case and weighted case. As an application, we obtain the…

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…

By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner-outer factorization. Here, a bounded analytic function is called \emph{inner} or \emph{outer} if multiplication by this…

Let $A = (A_1, \ldots, A_n)$ and $B = (B_1, \ldots, B_n)$ be row contractions on $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively, and $X$ be a row operator from $\oplus_{i=1}^n \mathcal{H}_2$ to $\mathcal{H}_1$. Let $D_{A^*} = (I - A…

Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…

The aim of this paper is to characterize in broad classes of means the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n)…

Nagel and Stein established $L^p$-boundedness for a class of singular integrals of NIS type, that is, non-isotropic smoothing operators of order 0, on spaces $\widetilde{M}=M_1\times...\times M_n,$ where each factor space $M_i, 1\leq i\leq…

We give an elementary construction of representing systems of the Cauchy kernels in the Hardy spaces $H^p$, $1 \le p <\infty$, as well as of representing systems of reproducing kernels in weighted Hardy spaces.

Spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure. We consider in this paper the reverse: for a positive matrix in the Hardy…

We construct a Schauder basis for the space $Hol(\mathbb D)$, the space of holomorphic functions on the closed unit disk, consisting entirely of finite Blaschke products. The expansion coefficients are given explicitly. Our result remains…

We consider Muckenhoupt weights $w$, and define weighted Hardy spaces $H^p_{\mathcal{T}}(w)$, where $\mathcal{T}$ denotes a conical square function or a non-tangential maximal function defined via the heat or the Poisson semigroup generated…

We consider the products of composition and differentiation operators on the Hardy space. We provide a complete characterization of the boundedness and compactness of these operators. Furthermore, we obtain the explicit condition for these…

Let $X$ be a ball quasi-Banach function space satisfying some mild assumptions and $H_X(\mathbb{R}^n)$ the Hardy space associated with $X$. In this article, the authors introduce both the Hardy space $H_X(\mathbb{R}^{n+1}_+)$ of harmonic…

In this paper, we investigate a class of fractional Hardy type operators $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ defined on higher-dimensional product spaces…

Let $(\mathcal{X},d,\mu)$ be a doubling metric measure space in the sense of R. R. Coifman and G. Weiss, $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ satisfying the Davies--Gaffney estimate, and $X(\mathcal{X})$ a ball…

Suppose that $(X,d,\mu)$ is a space of homogeneous type, with upper dimension $\mu$, in the sense of R. R. Coifman and G. Weiss. Let $\eta$ be the H\"{o}lder regularity index of wavelets constructed by P. Auscher and T. Hyt\"{o}nen. In this…

We study the behavior of Haar coefficients in Besov and Triebel-Lizorkin spaces on $\mathbb{R}$, for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to…

We prove a radial maximal function characterisation of the local atomic Hardy space h^1(M) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable…

According to Haar's Theorem, every compact group $G$ admits a unique (regular, right and) left-invariant Borel probability measure $\mu_G$. Let the Haar integral (of $G$) denote the functional $\int_G:\mathcal{C}(G)\ni f\mapsto \int…