Weighted Hardy Spaces Research Articles

Weighted Hardy factorizations in terms of Multilinear Calderón-Zygmund and fractional integral operators

The foundational paper of Coifman-Rochberg-Weiss provided a constructive proof of the weak factorizations of the classical Hardy space in terms of Riesz transforms. In this paper, we establish the factorization theorems for the weighted Hardy spaces. The results are also extended to the multilinear setting. Furthermore, we show that the weighted BMO space and weighted Lipschitz spaces are necessary for the weighted boundedness of commutators of the multilinear Calderón-Zygmund and fractional integral operators, respectively.

Weighted estimates for product singular integral operators in Journé’s class on RD-spaces

Abstract An RD-space 𝑀 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝑀. In this paper, firstly, the authors give the Plancherel–Pôlya characterization of product weighted Triebel–Lizorkin spaces and product weighted Besov spaces on RD-spaces and make some estimates for the product singular integral operators in Journé’s class on these function spaces. As a result of these conclusions, they present some sufficient conditions for the boundedness of product singular integral operators on the product Lipschitz spaces and product weighted Hardy spaces. Secondly, by the boundedness of lifting and projection operators, they also obtain that the dual spaces of the product weighted Hardy spaces are product weighted Carleson measure spaces. Using the idea of dual, the authors obtain the weighted boundedness of singular integral operators on the product weighted Carleson measure spaces.

Hyers–Ulam stabilities for [formula omitted]th differential operators on [formula omitted

In this paper, we introduce certain operators Tλm and Tφm on weighted Hardy spaces Hβ2, and in the following, we investigate their boundedness on Hβ2. After that, we prove the Hyers–Ulam stability for certain operators on weighted Hardy spaces Hβ2. Moreover, we show under what conditions these concepts are stable and unstable by using some examples. Finally, we investigate how differential operators are stable by using different sequences in a corollary and an example.

Boundedness for Commutators of Singular Integral Operator With Weaker Smooth Kernel on Weighted Hardy and Herz Spaces

In this job, let T be a variant of Hörmander’s singular operator and b ∈ Lipβ,ω(ℝn). We adopt the classic Harmonic analysis method and obtain that commutators are bounded on weighted Hardy space and weighted Herz space.

Bounded compact and dual compact approximation properties of Hardy spaces: New results and open problems

The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such spaces, to provide background for the open problems. Namely, we consider abstract Hardy spaces H[X(w)] built upon translation-invariant Banach function spaces X with weights w such that w∈X and w−1∈X′, where X′ is the associate space of X. We prove that if X is separable, then H[X(w)] has the BCAP with the approximation constant M(H[X(w)])≤2. Moreover, if X is reflexive, then H[X(w)] has the BCAP and the DCAP with the approximation constants M(H[X(w)])≤2 and M∗(H[X(w)])≤2, respectively. In the case of classical weighted Hardy space Hp(w)=H[Lp(w)] with 1<p<∞, one has a sharper result: M(Hp(w))≤2|1−2/p| and M∗(Hp(w))≤2|1−2/p|.

On the convergence of sequences of weighted composition operators on certain weighted Hardy spaces

On the convergence of sequences of weighted composition operators on certain weighted Hardy spaces

Characterizations of a class of weighted 𝐵𝑀𝑂 spaces via commutators of intrinsic square functions

The intrinsic square functions including the Lusin area function, Littlewood-Paley g g -function and g λ ∗ g^\ast _\lambda -function dominate pointwisely the classical Littlewood-Paley functions and can be used to characterize the weighted Hardy spaces and more general Musielak-Orlicz Hardy spaces etc. This paper shows that for b ∈ B M O ( R n ) b\in \mathrm {BMO(\mathbb {R}^n)} , the commutators generated by these intrinsic square functions with b b are bounded from H ω p ( R n ) H^p_\omega (\mathbb {R}^n) to L ω p ( R n ) L^p_\omega (\mathbb {R}^n) for some 0 &gt; p ≤ 1 0&gt;p\le 1 and ω ∈ A ∞ \omega \in A_\infty if and only if b ∈ B M O ω , p ( R n ) b\in \mathcal {BMO}_{\omega ,p}(\mathbb {R}^n) , which are a class of non-trivial subspaces of B M O ( R n ) \mathrm {BMO(\mathbb {R}^n)} .

Representations by Beurling Systems

We prove that a Beurling system with F∈Hp(D),1≤p&lt;∞ is an M—basis in Hp(D) with an explicit dual system. Any function f∈Hp(D),1≤p&lt;∞ can be expanded as a series by the system {zmF(z)}m=0∞. For different summation methods, we characterize the outer functions F for which the expansion with respect to the corresponding Beurling system converges to f. Related results for weighted Hardy spaces in the unit disc are studied. Particularly we prove Rosenblum’s hypothesis.

Weighted local Hardy spaces with variable exponents

AbstractThis paper defines local weighted Hardy spaces with variable exponents. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As applications, we obtain the boundedness of singular integral operators, the Littlewood–Paley characterization, and wavelet decomposition.

The commutators of multilinear Calderón–Zygmund operators on weighted Hardy spaces

In this paper, we study the behaviours of the commutators $[\vec b,\,T]$ generated by multilinear Calderón–Zygmund operators $T$ with $\vec b=(b_1,\,\ldots,\,b_m)\in L_{\rm loc}(\mathbb {R}^n)$ on weighted Hardy spaces. We show that for some $p_i\in (0,\,1]$ with $1/p=1/p_1+\cdots +1/p_m$ , $\omega \in A_\infty$ and $b_i\in \mathcal {BMO}_{\omega,p_i}$ ( $1\le i\le m$ ), which are a class of non-trivial subspaces of ${\rm BMO}$ , the commutators $[\vec b,\,T]$ are bounded from $H^{p_1}(\omega )\times \cdots \times H^{p_m}(\omega )$ to $L^p(\omega )$ . Meanwhile, we also establish the corresponding results for a class of maximal truncated multilinear commutators $T_{\vec b}^*$ .

Convergence analysis of approximation formulas for analytic functions via duality for potential energy minimization

We investigate the approximation formulas that were proposed by Tanaka & Sugihara (IMA J. Numer. Anal. 39(4):1957–1984, 2019), in weighted Hardy spaces, which are analytic function spaces with certain asymptotic decay. Under the criterion of minimum worst error of n-point approximation formulas, we demonstrate that the formulas are nearly optimal. We also obtain the upper bounds of the approximation errors that coincide with the existing heuristic bounds in asymptotic order by a duality theorem for the minimization problem of potential energy.

Representing Systems of Reproducing Kernels in Spaces of Analytic Functions

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.

Isomorphic Copies of ell ^infty in the Weighted Hardy Spaces on the Unit Disc

It is still unclear whether the density of analytic polynomials in an H-admissible space is sufficient to the minimality of the space? This question has a purely foundational background, relating fundamental concepts from the theory of H^p spaces. We hypothesize that there is no general relationship between the density of analytic polynomials and the R-admissibility of an H-admissible space. We solve this problem by finding suitable counterexamples of Hardy spaces built upon some weighted Lebesgue spaces. In particular, we provide a direct construction of weights from Szegő class, which guarantees the existence of isomorphic copies of the space of bounded sequences in weighted Hardy spaces on the unit disc.

Uncertainty inequality on weighted Hardy spaces

Abstract In this paper we introduce a weighted Hardy space ℋ β {\mathscr{H}_{\beta}} . This space generalizes some complex Hilbert spaces like the Dirichlet space 𝒟 {\mathscr{D}} , the Bergman space 𝒜 {\mathscr{A}} and the Segal–Bargmann space ℱ {\mathscr{F}} . It plays the role of background for our contribution. In particular, we study the derivative operator D and its adjoint operator L β {L_{\beta}} on ℋ β {\mathscr{H}_{\beta}} . Furthermore, we establish a general uncertainty inequality of Heisenberg type for the space ℋ β {\mathscr{H}_{\beta}} .

Weighted Composition Operators from the Bloch Spaces to Weighted Hardy Spaces on Bounded Symmetric Domains

Weighted Composition Operators from the Bloch Spaces to Weighted Hardy Spaces on Bounded Symmetric Domains

BOUNDEDNESS OF PSEUDO-DIFFERENTIAL OPERATORS ON WEIGHTED HARDY SPACES AND VARIABLE EXPONENTS HARDY LOCAL MORREY SPACES

In this paper, we use the atomic decomposition to establish the boundedness of pseudo-differential operators belonging to Hörmander class on weighted Hardy spaces $H^p(\omega)$ and on variable exponents Hardy local Morrey spaces $HLM_{p(\cdot)}^{u(\cdot)}$.

Weighted Composition Operators Between Weighted Hardy Spaces on Rooted Trees

In this paper, we introduce a discrete analogue of weighted Hardy spaces on rooted trees and study weighted composition operators between them in detail. In particular, we characterize bounded and compact weighted composition operators between discrete Hardy spaces.

The Commutators of Marcinkiewicz Integral Operators on Weighted Hardy Spaces

This paper is devoted to studying the behaviors of commutator μΩb generated by the Marcinkiewicz integral μΩ with b ∈ BMO(ℝn) on the weighted Hardy spaces. The authors show that there is a non-trivial subspace of BMO(ℝn), when b belongs to this subspaces, such that μΩb is bounded from the weighted Hardy spaces H (ℝn) to the weighted Lebesgue spaces L (ℝn) for some 0 < p ≤ 1.

Weighted estimates for commutators associated to singular integral operator satisfying a variant of Hörmander's condition

&lt;abstract&gt;&lt;p&gt;In this paper, we establish some boundedness for commutators generated by the singular integral operator satisfying a variant of Hörmander's condition and a weighted BMO function on weighted Hardy spaces and weighted Herz spaces. As an application, we obtain some classical results.&lt;/p&gt;&lt;/abstract&gt;

Generalized weighted composition operators on weighted Hardy spaces

Generalized weighted composition operators on weighted Hardy spaces