Product 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

Hardy–Littlewood-type theorems for Fourier transforms in [formula omitted

We obtain Fourier inequalities in the weighted Lp spaces for any 1<p<∞ involving the Hardy–Cesàro and Hardy–Bellman operators. We extend these results to product Hardy spaces for p⩽1. Moreover, boundedness of the Hardy-Cesàro and Hardy-Bellman operators in various spaces (Lebesgue, Hardy, BMO) is discussed. One of our main tools is an appropriate version of the Hardy–Littlewood–Paley inequality ‖fˆ‖Lp′,q≲‖f‖Lp,q.

Characterizations of Product Hardy Spaces in Bessel Setting

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt–Stein, especially the generalised Cauchy–Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the generalised Cauchy–Riemann equations, the extension of Merryfield’s result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators.

Weighted product Hardy space associated with operators

Assuming that the operators L1, L2 are self-adjoint and $${{\rm{e}}^{ - t{L_i}}}$$ (i = 1, 2) satisfy the generalized Davies-Gaffney estimates, we shall prove that the weighted Hardy space $$H_{{L_1},{L_2},\omega }^1({\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}})$$ associated to operators L1, L2 on product domain, which is defined in terms of area function, has an atomic decomposition for some weight ω.

Hardy space estimates for bi-parameter Littlewood-Paley square functions

Suppose that g(f) are bi-parameter Littlewood-Paley square functions which were introduced by H. Martikainen. It is known that the L2(ℝn × ℝm) boundedness and the H1 (ℝn × ℝm)-L1(ℝn × ℝm) boundedness of g(f) have been proved by H. Martikainen and by Z. Li and Q. Xue, respectively. In this paper, we apply the vector-valued theory, the atomic decomposition of product Hardy spaces, and Journe’s covering lemma to show that g(f) are bounded from Hp(ℝn × ℝm) to Lp(ℝn × ℝm) with p smaller than 1.

Characterizations of product Hardy space associated to Schrödinger operators

Let L1 and L2 be the Schrodinger operators on ℝn and ℝm, respectively. By using different maximal functions and Littlewood-Paley g function on distinct variables, in this paper, some characterizations for functions in the product Hardy space $$H_{{L_1},{L_2}}^1$$(ℝn × ℝm) associated to operators L1 and L2 are obtained.

Relations Between Product and Flag Hardy Spaces

In Nagel et al. (J Funct Anal 181:29–118, 2001), Nagel–Ricci–Stein established the relationships between product kernels and flag kernels on the Euclidean space, that is, product kernels are finite sums of flag kernels. The main purpose of this paper is to characterize the product Hardy space as the intersection of flag Hardy spaces, and characterize the product Carleson measure space as the sum of flag spaces.

Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators

Let $L_1$ and $L_2$ be nonnegative self-adjoint operators acting on $L^2(X_1)$ and $L^2(X_2)$, respectively, where $X_1$ and $X_2$ are spaces of homogeneous type. Assume that $L_1$ and $L_2$ have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ associated to $L_{1}$ and $L_{2}$, for $p \in (0, \infty)$ and the weight $w$ belongs to the product Muckenhoupt class $ A_{\infty}(X_{1} \times X_{2})$. Our main result is that the spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ introduced via area functions can be equivalently characterized by the Littlewood–Paley $g$-functions and $g^{\ast}_{\lambda_{1}, \lambda_{2}}$-functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of $L_1$ and $L_2$. Our results are new even in the unweighted product setting.

Boundedness of bi-parameter Littlewood–Paley operators on product Hardy space

Let $$n_1=n\ge 1, n_2=m\ge 1$$ and $$\lambda _2>1$$ . For any $$x=(x_1,x_2) \in \mathbb {R}^n\times \mathbb {R}^m$$ , let g and $$g_{\mathbf {\lambda }}^*$$ be the bi-parameter Littlewood–Paley square functions defined by $$\begin{aligned} g(f)(x)&= \left( \int _0^{\infty }\int _0^{\infty }|\theta _{t_1,t_2} f(x_1,x_2)|^2 \frac{dt_1}{t_1} \frac{dt_2}{t_2} \right) ^{1/2}, \hbox { and} g_{\mathbf {\lambda }}^*(f)(x)&= \left( \iint _{\mathbb {R}^{m+1}_{+}} \iint _{\mathbb {R}^{n+1}_{+}} \prod _{i=1}^2\Big (\frac{t_1}{t_i + |x_i - y_i|}\Big )^{n_i \lambda _i}\right. \\&\left. \quad \times \, |\theta _{t_1,t_2} f(y_1,y_2)|^2 \frac{dy_1 dt_1}{t_1^{n+1}} \frac{dy_2 dt_2}{t_2^{m+1}} \right) ^{1/2}, \end{aligned}$$ where $$\theta _{t_1,t_2} f(x_1, x_2) = \iint _{\mathbb {R}^n\times \mathbb {R}^m} s_{t_1,t_2}(x_1,x_2,y_1,y_2)f(y_1,y_2) dy_1dy_2$$ . It is known that the $$L^2$$ boundedness of bi-parameter g and $$g_{\mathbf {\lambda }}^*$$ have been established recently by Martikainen, and Cao, Xue, respectively. In this paper, under certain structural conditions assumed on the kernel $$s_{t_1,t_2},$$ we show that both g and $$g_{\mathbf {\lambda }}^*$$ are bounded from product Hardy space $$H^1(\mathbb {R}^n\times \mathbb {R}^m)$$ to $$L^1(\mathbb {R}^n\times \mathbb {R}^m)$$ . As consequences, the $$L^p$$ boundedness of g and $$g_{\mathbf {\lambda }}^*$$ will be obtained for $$1<p<2$$ .

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let $$\vec A: = \left( {{A_1},{A_2}} \right)$$ be a pair of expansive dilations and φ: ℝ n ×ℝ m ×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ via the anisotropic Lusin-area function and establish its atomic characterization, the $$\vec g$$ -function characterization, the $$\vec g_\lambda ^*$$ -function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet ( $$\left( {\varphi ,q,\vec s} \right)$$ ), if T is a sublinear operator and maps all ( $$\left( {\varphi ,q,\vec s} \right)$$ )-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to L φ (R n × R m ) and from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to itself, whose kernels are adapted to the action of $$\vec A$$ . The results of this article essentially extend the existing results for weighted product Hardy spaces on ℝ n × ℝ m and are new even for classical product Orlicz-Hardy spaces.

Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels

Let \(T_{\vec{b}}\) and \(T_{\Pi b}\) be the commutators in the jth entry and iterated commutators of the multilinear Calderon-Zygmund operators, respectively. It was well known that the commutators of linear Calderon-Zygmund operators were not of weak type \((1,1)\) and \((H^{1}, L^{1})\), but they did satisfy certain endpoint \(L\log L\) type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that \(T_{\vec{b}}\) and \(T_{\Pi b}\) are bounded from the product Hardy space \(H^{1}\times\cdots\times H^{1}\) to weak \(L^{\frac{1}{m},\infty}\) space, whenever the kernel satisfies a class of Dini type condition. This was done by using a key lemma given by Christ, a very complex decomposition of the integrand domains, and carefully splitting the commutators into several terms.

Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases

In this paper, we first show that the remarkable orthonormal wavelet expansion for L p constructed recently by Auscher and Hytönen also converges in certain spaces of test functions and distributions. Hence we establish the theory of product Hardy spaces on spaces X ˜ = X 1 × X 2 × ⋅ ⋅ ⋅ × X n , where each factor X i is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood–Paley theory on X ˜ , which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this paper includes product H p , the dual CMO p of H p with the special case BMO = CMO 1 , and the predual VMO of H 1 . We also use the wavelet expansion to establish the Calderón–Zygmund decomposition for product H p , and deduce an interpolation theorem. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood–Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing wavelet analysis on spaces of homogeneous type.

Marstrand's Approximate Independence of Sets and Strong Differentiation of the Integral

A constructive proof is given for the existence of a function belonging to the product Hardy space $H^1(\mathsf{R} \times \mathsf{R})$ and the Orlicz space $L(\log L)^{\epsilon}(\mathsf{R}^{2})$ for all $0&lt;\epsilon &lt;1$, for all whose integral is not strongly differentiable almost everywhere on a set of positive measure. It consists of a modification of a non-negative function created by J. M. Marstrand. In addition, we generalize the claim concerning "approximately independent sets" that appears in his work in relation to hyperbolic-crosses. Our generalization, which holds for any sets with boundary of sufficiently low complexity in any Euclidean space, has a version of the second Borel-Cantelli Lemma as a corollary.

On weak-star convergence in product Hardy spaces on spaces of homogeneous type

A classical theorem of Jones and Journe on weak-star convergence in the Hardy space $H^1$ was generalised to the multiparameter setting by Pipher and Treil. We prove the analogous result when the underlying space is a product space of homogeneous type. The main tools we use for this setting are from recent work in papers by Chen, Li and Ward and by Han, Li and Ward.

Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type

The aim of this article is to develop the theory of product Hardy spaces associated with operators which possess the weak assumption of Davies–Gaffney heat kernel estimates, in the setting of spaces of homogeneous type. We also establish a Calderon–Zygmund decomposition on product spaces, which is of independent and use it to study the interpolation of these product Hardy spaces. We then show that under the assumption of generalized Gaussian estimates, the product Hardy spaces coincide with the Lebesgue spaces, for an appropriate range of p.

Dyadic structure theorems for multiparameter function spaces

We prove that the multiparameter (product) space BMO of functions of bounded mean oscillation can be written as the intersection of finitely many dyadic product BMO spaces, with equivalent norms, generalizing the one-parameter result of T. Mei. We establish the analogous dyadic structure theorems for the space VMO of functions of vanishing mean oscillation, for A_p weights, for reverse-Hölder weights and for doubling weights. We survey several definitions of VMO and prove their equivalences, in the continuous, dyadic, one-parameter and product cases. In particular, we introduce the space of dyadic product VMO functions. We show that the weighted product Hardy space H^1_{\omega} is the sum of finitely many translates of dyadic weighted H^1_{\omega} , for each A_{\infty} weight \omega , and that the weighted strong maximal function is pointwise comparable to the sum of finitely many dyadic weighted strong maximal functions, for each doubling weight \omega . Our results hold in both the compact and non-compact cases.

Various characterizations of product Hardy spaces associated to Schrödinger operators

For every i = 1, 2, we let \(L_i = - \Delta _{n_i } + V_i\) be a Schrodinger operator on \(\mathbb{R}^{n_i }\) in which \(V_i \in L_{loc}^1 (\mathbb{R}^{n_i } )\) is a non-negative function on \(\mathbb{R}^{n_i }\). We obtain some characterizations for functions in the product Hardy space \(H_{L_1 ,L_2 }^1 (\mathbb{R}^{n_1 } \times \mathbb{R}^{n_2 } )\) associated to L1 and L2 by using different norms on distinct variables.

Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group

Marcinkiewicz multipliers are L^{p} bounded for 1<p<\infty on the Heisenberg group H^{n}\simeqC^{n}\timesR (D. Muller, F. Ricci and E. M. Stein) despite the lack of a two parameter group of automorphic dilations on H^{n}. This lack of dilations underlies the inability of classical one or two parameter Hardy space theory to handle Marcinkiewicz multipliers on H^{n} when 0<p\leq1. We address this deficiency by developing a theory of flag Hardy spaces H_{flag}^{p} on the Heisenberg group, 0<p\leq1, that is in a sense `intermediate' between the classical Hardy spaces H^{p} and the product Hardy spaces H_{product}^{p} on C^{n}\timesR. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on H_{flag}^{p}, as well as from H_{flag}^{p} to L^{p}, for 0<p\leq1. We characterize the dual spaces of H_{flag}^{1} and H_{flag}^{p}, and establish a Calder\'on-Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces H_{flag}^{p}. In particular, this recovers the L^{p} results by interpolating between those for H_{flag}^{p} and L^{2} (but regularity sharpness is lost).

Differentiation of the integral, Hardy spaces and Calderón–Zygmund operators in the product setting

This article concerns strong differentiation and operators on product Hardy spaces. We first show that an example, created by Papoulis, of a function on R2 whose integral is strongly differentiable almost everywhere, but the integral of its absolute value fails to be strongly differentiable on a set of positive measure, belongs to the product Hardy space H1(R×R). The methods that we develop enable us to present a relaxed version of Chang–Fefferman p-atoms with a lower number of required vanishing moments and no smoothness needed on the elementary particles. In analogy with the proof of this result, we show a generalization of a theorem of R. Fefferman which concludes Hp→Lp, 0<p≤1, boundedness of multiparameter operators from their behavior on rectangle atoms. In addition, we extend a result of Pipher concerning boundedness of multiparameter Calderón–Zygmund operators from Hp to Lp.

Boundedness of Calderon-Zygmund operators on weighted product Hardy spaces

Boundedness of Calderon-Zygmund operators on weighted product Hardy spaces