We study the boundedness of inhomogeneous Journ & eacute;'s operators on multi-parameter local Hardy spaces. Recently, a class of local multi-parameter paraproducts was shown to obstruct particular T1-type theorems in this context. In this paper, we study the boundedness of the adjoints of these local multi-parameter paraproducts. The theory is more challenging due to the lack of vanishing moments. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

It is known that the pseudodifferential operator is an important tool to study hypo-elliptic equations. The known results include the continuity properties of pseudodifferential operators on local Hardy spaces h^p for 0

Building upon the conditions satisfied by the double Hilbert transform, Journé introduced a class of non-convolution-type singular integral operators with product structure, now referred to as the Journé class. To investigate the boundedness of operators on multi-parameter local Hardy spaces, researchers introduced inhomogeneous Journé-type singular integral operators. Subsequent studies have demonstrated that many operators naturally fall within this inhomogeneous Journé class. Analogous to the two aforementioned classes of sets, mixed Journé class were introduced. Up to now, the question of whether the set of mixed Journé class is non-empty remains open. The primary objective of this study is to provide an affirmative response to this question.

It is well known that an pseudodifferential operator T-sigma with sigma is an element of S-rho,delta(m)(R-n) is bounded on L-2(R-n) if m <= 0 , m + n/2(delta - rho) < 0 which was established by Hormander. In this paper, we generalize this result to multi-parameter pseudodifferential operators. Using the atomic decomposition theory, we obtain the boundedness from h(p)(R-n1 x R-n2) to L-p(Rn1+n2) of bi-parameter pseudodifferential operators for 0 < p <= 1. Calderon-Zygmund decomposition and interpolation theory on h(p)(R-n1 x R-n2) are also established.

To study the boundedness of operators on multi-parameter local Hardy space h^p(ℝ^n_1×ℝ^n_2) , inhomogeneous Journé class has been introduced. It is well known that operators in the Journé class are bounded on multi-parameter Hardy space H^p(ℝ^n_1×ℝ^n_2) if and only if T^*_1(1)=T^*_2(1)=0 for p near 1. Under the same conditions, operators in inhomogeneous Journé class are bounded on h^p(ℝ^n_1×ℝ^n_2) . In this paper, We give an operator belonging to the inhomogeneous Journé class without T^*_1(1)=T^*_2(1)=0 and prove its boundedness from h^p(ℝ^n_1×ℝ^n_2) to h^p(ℝ^n_1×ℝ^n_2) by almost orthogonality estimates. It implies that T^*_1(1)=T^*_2(1)=0 is not a necessary condition for the boundedness on h^p(ℝ^n_1×ℝ^n_2) of a singular operator in the inhomogeneous Journé class.

In this paper, we discuss the boundedness of mixed Journé's class operators on weighted multi-parameter mixed Hardy spaces via atoms decomposition. Moreover, we give a specific singular integral operator in mixed Journé's class which has better properties.

It is well-known that the pseudodifferential operator with the symbol in Bony class, a subset of S_1,1^0(ℝ^n) , is bounded on L^2(ℝ^n) . The main purpose of this paper is to extend the classical results to multi-parameter case, i.e., to discuss the boundedness on L^2(ℝ^n_1+n_2) and on h^p(ℝ^n_1×ℝ^n_2) (0

Though atomic decomposition is a very useful tool for studying the boundedness on Hardy spaces for some sublinear operators, untill now, the boundedness of operators on weighted Hardy spaces in a multi-parameter setting has been established only by almost orthogonality estimates. In this paper, we mainly establish the boundedness on weighted multi-parameter local Hardy spaces via atomic decomposition.

It is well-known that atomic decomposition is an important tool to study the boundedness of some singular integral operators on Hardy spaces. Moreover, to study the boundedness of an operator in the Journé class, Fefferman R. builded a criterion by considering its action on rectangle atoms only. In this paper, we mainly establish atomic decomposition of multi-parameter mixed Hardy space which has been developed recently.

Applying discrete Calderón’s identity, we study weighted multi-parameter mixed Hardy space Hmixp(ω,ℝn1×ℝn2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{{\rm{mix}}}^p(\omega ,{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}})$$\end{document}. Different from classical multi-parameter Hardy space, this space has characteristics of local Hardy space and Hardy space in different directions, respectively. As applications, we discuss the boundedness on Hmixp(ω,ℝn1×ℝn2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{{\rm{mix}}}^p(\omega ,{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}})$$\end{document} of operators in mixed Journé’s class.

To study the boundedness of bi-parameter singular integral operators of non-convolution type in the Journe class, Fefferman discovered a boundedness criterion on bi-parameter Hardy spaces H-p(R-n1 x R-n2) by considering the action of the operators on rectangle atoms. More recently, the theory of multiparameter local Hardy spaces has been developed by the authors. In this paper, we establish this type of boundedness criterion on weighted bi-parameter local Hardy spaces h(omega)(p) (R-n1 x R-n2). In comparison with the unweighted case, the uniform boundedness of rectangle atoms on weighted local bi-parameter Hardy spaces, which is crucial to establish the atomic decomposition on bi-parameter weighted local Hardy spaces, is considerably more involved. As an application, we establish the boundedness of bi-parameter pseudodifferential operators, including h(omega)(p)(R-n1 x R-n2) to L-omega(p)(Rn1+n2) and h(omega)(p)(R(n1 )x R-n2) to h(omega)p(R-n1 x R-n2) for all 0 < p <= 1, which sharpens our earlier result even in the unweighted case requiring max{n(1)/n(1) + 1, n(2)/n(2) + 1} < p <= 1.

In this paper, we study the duality theory of the multiparameter local Hardy spaces hpℝn1×ℝn2 , and we prove that hpℝn1×ℝn2∗=cmopℝn1×ℝn2 , where cmopℝn1×ℝn2 are defined by discrete Carleson measure. Moreover, we discuss the relationship among cmopℝn1×ℝn2 , Lippℝn1×ℝn2 , and rectangle cmorectpℝn1×ℝn2 .

In this paper, we prove that $$cmo^{p}(\mathbb {R}^{n})$$ and $$\Lambda _{n(\frac{1}{p}-1)}$$ , the dual spaces of local Hardy space $$h^{p}(\mathbb {R}^{n})$$ , are coincide with equivalent norms for $$\frac{n}{n+1}<p\le 1$$ . Moreover, this space can be characterized by another simple norm. As an application, we prove the $$h^{p}(\mathbb {R}^{n})$$ boundedness of inhomogeneous para-product operators.

The classical theory of one-parameter harmonic analysis may be considered as centering around the HardyLittlewood maximal operator and its relationship with certain singular integral operators which commute with the usual one-parameter dilations on Rm, given by δðxÞ = ðδx1, ⋯,δxmÞ, δ > 0. If this isotropic dilation is replaced by more general nonisotropic groups of dilations, then many nonisotropic variants of the classical theories can be produced, such as the strong maximal functions and multiparameter singular integral operators, corresponding to the multiparameter dilations δ : x⟶ ðδ1x1, δ2x2Þ, x = ðx1, x2Þ ∈Rn ×Rm, and δ = ðδ1, δ2Þ, δ1, δ2 > 0. Such a multiparameter theory has been developed extensively over the past decades. We refer the reader to the work in [1–18]. Since HpðRnÞ space is well suited only to the pure Fourier analysis and not stable under multiplication by test functions which associated with PDE, in [19], Goldberg introduced the class of localizable Hardy spaces hðRnÞ, 0 < p <∞. Let φ ∈ SðRnÞ with Ð φ ≠ 0 and

In this paper, we establish a boundedness criterion of a class of inhomogeneous Journe's type of multi-parameter singular integral operators on multi-parameter local Hardy spaces h(p)(R-n1 x center dot center dot center dot x R-nk) recently introduced in Ding et al. (2019). The consideration of such inhomogeneous Journe's type of multi-parameter singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces. The lack of the vanishing moments of both the corresponding functions in the multi-parameter local Calderon reproducing formula and the multi-parameter local Littlewood-Paley-Stein functions, as well as the lack of vanishing moments of atoms in the multi-parameter local Hardy spaces h(p)(R-n1 x center dot center dot center dot xR(nk)) with large size all create substantial difficulties in our proofs. (C) 2020 Elsevier Ltd. All rights reserved.

In this paper, we prove that cmo^p(ℝ^n) and Λ _n(1/p-1) , the dual spaces of local Hardy space h^p(ℝ^n) , are coincide with equivalent norms for n/n+1<p≤ 1 . Moreover, this space can be characterized by another simple norm. As an application, we prove the h^p(ℝ^n) boundedness of inhomogeneous para-product operators.

Multi-parameter mixed Hardy space $$H_{{\rm{mix}}}^p$$ is introduced by a new discrete Calderón’s identity. As an application, we obtain the $$H_{{\rm{mix}}}^p \to {L^p}\left({{ℝ^{{n_1} + {n_2}}}} \right)$$ boundedness of operators in the mixed Journé’s class.

After the celebrated work of L. Hormander 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 partial derivative 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 pseudodifferential operators are L-p(R-n) bounded for 1 < p < infinity, but only bounded on local Hardy spaces h(p)(R-n) introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27- 42] for 0 < p <= 1. Though much work has been done on the L-p(R-n(1) x R-n(2)) boundedness for 1 < p < infinity and Hardy H-p (R-n(1) x R-n2) boundedness for 0 < p <= 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. The main purpose of this paper is to establish the boundedness of multiparameter pseudo-differential operators on multi-parameter local Hardy spaces h(p)(R-n1 x R-n2 ) for 0 < p <= 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].

The boundedness of operators on Hardy spaces is usually given by atomic decomposition. In this paper, we obtain the boundedness of singular integral operators in mixed Journe class on mixed Hardy spaces by a direct method.

The purpose of this paper is to provide necessary and sufficient conditions of the boundedness for singular integrals on the local Hardy space and its dual. Particularly the singular integrals considered in this paper include the pseudo-differential operators $T_{\sigma }f(x)=\int \limits \sigma (x \xi )e^{2\pi ix\xi }\hat {f}(\xi )d\xi $ with $\sigma \in S_{1 0}^{0}$ . As a consequence our results give another proof of the boundedness of the pseudo-differential operators on the local Hardy space (Goldberg Duke Math. J. 46(1) 27–42 1979).