Real Hardy Space Research Articles

Variation type characterization of product Hardy spaces

In the multidimensional Euclidean space, except the classical real Hardy space, there are numerous product ones. We associate with each of them a class of functions related to the known variations and new ones. Such a characterization is fulfilled by means of the integrability of the Fourier transform

Hausdorff operators on compact abelian groups

AbstractNecessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces , real Hardy space , , and for compact Abelian group G. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on G and examples are also considered.

HAUSDORFF OPERATORS OVER DOUBLE COSET SPACES OF GROUPS WITH LOCALLY DOUBLING PROPERTY

We consider the boundedness of Hausdorff operators on the real Hardy spaces $$H^{1}$$ over double coset spaces of locally compact groups with the local doubling property and approximate midpoint property. The $$L^{q}$$ spaces as well as the example of the group of Euclidean motions are also considered.

Hausdorff operators on Hardy type spaces

During last 20 years, an essential part of the theory of Hausdorff operators is concentrated on their boundedness on the real Hardy space 𝐻1(R𝑑). The spaces introduced by Sweezy are, in many respects, natural extensions of this space. They are nested in full between 𝐻1(R𝑑) and 𝐿10 (R𝑑). Contrary to 𝐻1(R𝑑), they are subject only to atomic characterization. For the estimates of Hausdorff operators on 𝐻1(R𝑑), other characterizations have always been applied. Since this option is excluded in the case of Sweezy spaces, in this paper an approach to the estimates of Hausdorff operators is elaborated, where only atomic decompositions are used. While on 𝐻1(R𝑑) this approach is applicable to the atoms of the same type, on the Sweezy spaces the same approach is not less effective for the sums of atoms of various types. For a single Hausdorff operator, the boundedness condition does not depend on the space but only on the parameters of the operator itself. The space on which this operator acts is characterized by the choice of atoms. An example is given (two-dimensional, for simplicity), where a matrix dilates the argument only in one variable.

Real Hardy Space, Multidimensional Variations, and Integrability of the Fourier Transform

A new class of functions is introduced closely related to that of functions with bounded Tonelli variation and to the real Hardy space. For this class, conditions for integrability of the Fourier transform are established.

Hausdorff operators on homogeneous spaces of locally compact groups

Hausdorff operators on the real line and multidimensional Euclidean spaces originated from some classical summation methods. Now it is an active research area. Hausdorff operators on general groups were defined and studied by the author since 2019. The purpose of this paper is to define and study Hausdorff operators on Lebesgue and real Hardy spaces over homogeneous spaces of locally compact groups. We introduce in particular an atomic Hardy space over homogeneous spaces of locally compact groups and obtain conditions for boundedness of Hausdorff operators on such spaces. Several corollaries are considered and unsolved problems are formulated.

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

Abstract After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} bounded for 1 < p < ∞ {1<p<\infty} , but only bounded on local Hardy spaces h p ⁢ ( ℝ n ) {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0 < p ≤ 1 {0<p\leq 1} . Though much work has been done on the L p ⁢ ( ℝ n 1 × ℝ n 2 ) {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 1 < p < ∞ {1<p<\infty} and Hardy H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 0 < p ≤ 1 {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of 0 < p ≤ 1 {0<p\leq 1} . The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for 0 < p ≤ 1 {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

Triebel–Lizorkin space estimates for evolution equations with structure dissipation

In this paper, we are concerned with the generalized wave equations in the homogeneous Triebel–Lizorkin spaces. The long time decay estimates are obtained by the decomposition of the unit, duality property and the multiplier theorems, extending the known results for the generalized wave equations with structure dissipation in the real Hardy spaces.

Commutators, BMO, Hardy Spaces and Factorization: A Survey

In this survey we discuss the connection between commutator operators and functions of bounded mean oscillation, and at the same time outline the parallel story of the real Hardy space and weak factorization of these functions. We provide motivation as to why these questions are interesting and highlight the many different methods of proof that exist for proving results of the type discussed.

Estimates of Damped Fractional Wave Equations

Abstract In this paper, for the high frequency part of the solution u(x, t) to the linear fractional damped wave equation, we derive asymptotic-in-time linear estimates in Triebel-Lizorkin spaces. Thus we obtain long time decay estimates in real Hardy spaces Hp for u(x, t). The obtained results are natural extension of the known Lp estimates. Our proof is based on some basic properties of the Triebel-Lizorkin space, as well as an atomic decomposition introduced by Han, Paluszynski and Weiss.

Addendum to “Boundedness of Hausdorff operators on real Hardy spaces H1 over locally compact groups” [J. Math. Anal. Appl. 473 (2019) 519–533

Addendum to “Boundedness of Hausdorff operators on real Hardy spaces H1 over locally compact groups” [J. Math. Anal. Appl. 473 (2019) 519–533

The Dunkl-Hausdorff operator is bounded on the real Hardy space

ABSTRACTLiflyand and Móricz in Liflyand and Móricz [The Hausdorff operator is bounded on the real Hardy space . Proc Amer Math Soc. 1999;128:1391–1396; Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces. Acta Math Hungar. 2002;97(1–2):133–143] studied the Hausdorff operator on the real Hardy space and proved the boundedness property and the commuting relations of this operator and Hilbert transformations. The main objective of this paper is extending these results to the context of Dunkl theory.

Harmonic Functions, Conjugate Harmonic Functions and the Hardy Space H^1 in the Rational Dunkl Setting

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 consider systems of conjugate (partial _t^2+Delta _{mathbf {x}})-harmonic functions satisfying an appropriate uniform L^1 condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space H^1_{Delta }, can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood–Paley square functions.

Boundedness of Hausdorff operators on real Hardy spaces H1 over locally compact groups

Results of Liflyand and collaborators on the boundedness of Hausdorff operators on the Hardy space H 1 over finite-dimensional real space are generalized to the case of locally compact groups that are spaces of homogeneous type. Special cases and examples of compact Lie groups, homogeneous groups (in particular the Heisenberg group) and finite-dimensional spaces over division rings are considered. In conclusion, we solve for the space L 2 an open question on compactness of a Hausdorff operator posed by Liflyand.

Schrödinger operators involving singular potentials and measure data

We study the existence of solutions of the Dirichlet problem for the Schrödinger operator with measure data{−Δu+Vu=μin Ω,u=0on ∂Ω. We characterize the finite measures μ for which this problem has a solution for every nonnegative potential V in the Lebesgue space Lp(Ω) with 1≤p≤N2. The full answer can be expressed in terms of the W2,p capacity for p>1, and the W1,2 (or Newtonian) capacity for p=1. We then prove the existence of a solution of the problem above when V belongs to the real Hardy space H1(Ω) and μ is diffuse with respect to the W2,1 capacity.

Fourier multipliers on the real Hardy spaces

We provide a variant of Hytonen’s embedding theorem, which allows us to extend and unify several sufficient conditions for a function to be a Fourier multiplier on the real Hardy spaces.

The fractional Hausdorff operators on the Hardy spaces H p (ℝ n )

In this paper, we study the high-dimensional fractional Hausdorff operators and establish their boundedness on the real Hardy spaces Hp(ℝn) for 0 < p < 1.

Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation

In this paper we derive asymptotic-in-time linear estimates in Hardy spaces \(H^p(\mathbb {R}^{n})\) for the Cauchy problem for evolution operators with structural dissipation. The obtained estimates are a natural extension of the known \(L^p-L^q\) estimates, \(1\le p\le q\le \infty \), for these models. Different, standard, tools to work in Hardy spaces, are used to derive optimal estimates.

Some remarks on K-closedness for the couples of real Hardy spaces

In this paper K-closedness is proved in the case of the couple of real Hardy spaces (Hr(Rn),Hp(Rn)) in the corresponding couple of Lebesgue spaces for n−1n<r<p≤∞. This means roughly that any measurable decomposition of an analytic function gives rise to an “analytic” decomposition with summands of roughly the same size. The proof uses Bourgain's method, the atomic decomposition for Hardy spaces and the subharmonic property of the gradient of a system of conjugate harmonic functions.

Trigonometric multipliers on real periodic Hardy spaces

In this paper we are concerned with the boundedness of Hormander–Mihlin multipliers of order $$\,r\, (1\le r<\infty )\,$$ on the real periodic Hardy space $$\,\mathcal H^p_{2\pi }, 0<p<1.$$ The case $$\,r=1\,$$ corresponds to the classical Marcinkiewicz multiplier condition which is known to be sufficient for the boundedness of the trigonometric multipliers on the Lebesgue spaces $$L_{2\pi }^p, 1<p<\infty ,$$ but not on $$L^1_{2\pi }.$$ Daly (Can Math Bull 48:370–381, 2005) and the author showed among others that this is the situation for the Hormander–Mihlin condition with $$\,r>1.$$ On the other hand the boundedness extends to $$\,\mathcal H_{2\pi }$$ if $$\,r>1,$$ but not if $$\,r=1.$$ Generalizing this result in the present paper we show that the scale of Hardy spaces $$\,\mathcal H^p_{2\pi }\, (0<p<1)\,$$ is more adequate than that of the Lebesgue spaces in this regard. Namely, for any $$\,r>1\,$$ we give a sharp bound $$\,p_r<1\,$$ such that if $$\,p>p_r\,$$ then the Hormander–Mihlin condition of order $$\,r\,$$ is sufficient on $$\mathcal H^p_{2\pi }$$ .