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Singular Integrals and Weighted Triebel-Lizorkin and Besov Spaces of Arbitrary Number of Parameters 被引量:7
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作者 guo zhen lu Yue Ping ZHU 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2013年第1期39-52,共14页
Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the... Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon's identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe's covering lemma in multiparameter setting. 展开更多
关键词 Singular integrals multiparameter weighted Triebel-Lizorkin spaces multiparameter weighted Besov spaces discrete Littlewood-Paley analysis discrete Calderon identity vector-valued maximal functions
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Atomic Decomposition and Boundedness Criterion of Operators on Multi-parameter Hardy Spaces of Homogeneous Type 被引量:3
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作者 guo zhen lu Ya Yuan XIAO 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2012年第7期1329-1346,共18页
The main purpose of this paper is to derive a new (p,q)-atomic decomposition on the multi-parameter Hardy space HP(X1 × X2) for 0 〈 po 〈 P ≤ 1 for some po and all 1 〈 q 〈 ∞, where X1 ×X2 is the pro... The main purpose of this paper is to derive a new (p,q)-atomic decomposition on the multi-parameter Hardy space HP(X1 × X2) for 0 〈 po 〈 P ≤ 1 for some po and all 1 〈 q 〈 ∞, where X1 ×X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both L^q(X1 × X2) (for 1 〈 q 〈 ∞) and Hardy space HP(X1× X2) (for 0 〈 p _〈 1). As an application, we prove that an operator T, which is bounded on Lq(X1× X2) for some 1 〈 q 〈 ∞, is bounded from H^p(X1 × X2) to L^p(X1 × X2) if and only if T is bounded uniformly on all (p, q)-product atoms in LP(X1 × X2). The similar boundedness criterion from HP(X1 × X2) to HP(X1 × X2) is also obtained. 展开更多
关键词 Atomic decomposition boundedness criterion spaces of homogeneous type multi- parameter Hardy spaces CalderSn's reproducing formula Littlewood Paley theory Journ's covering lemma
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Multi-parameter Triebel-Lizorkin and Besov Spaces Associated with Flag Singular Integrals 被引量:4
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作者 Yong DING guo zhen lu Bo Lin MA 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2010年第4期603-620,共18页
Though the theory of one-parameter Triebel-Lizorkin and Besov spaces has been very well developed in the past decades, the multi-parameter counterpart of such a theory is still absent. The main purpose of this paper i... Though the theory of one-parameter Triebel-Lizorkin and Besov spaces has been very well developed in the past decades, the multi-parameter counterpart of such a theory is still absent. The main purpose of this paper is to develop a theory of multi-parameter Triebel-Lizorkin and Besov spaces using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter structure. It is motivated by the recent work of Han and Lu in which they established a satisfactory theory of multi-parameter Littlewood-Paley-Stein analysis and Hardy spaces associated with the flag singular integral operators studied by Muller-Ricci-Stein and Nagel-Ricci-Stein. We also prove the boundedness of flag singular integral operators on Triebel-Lizorkin space and Besov space. Our methods here can be applied to develop easily the theory of multi-parameter Triebel-Lizorkin and Besov spaces in the pure product setting. 展开更多
关键词 flag singular integrals multiparameter Triebel-Lizorkin spaces discrete Calderdn repro- ducing formulas discrete Littlewood-Paley-Stein analysis
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Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces 被引量:2
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作者 Xia LI guo zhen lu Han Li TANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2015年第7期1067-1085,共19页
In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebe... In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander's condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander's condition, but they also hold for Grushin vector fields as well with obvious modifications. 展开更多
关键词 Poincare inequalities the representation formula fractional integrals on homogeneousspaces vector fields satisfying Hormander's condition stratified groups high order non-isotropic Sobolev spaces with variable exponents Sobolev inequalities with variable exponents
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Global Poincaré Inequalities on the Heisenberg Group and Applications 被引量:1
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作者 Yu Xin DONG guo zhen lu Li Jing SUN 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2007年第4期735-744,共10页
Let f be in the localized nonisotropic Sobolev space Wloc^1,p (H^n) on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n + 2 is the homogeneous dimension of H^n. Suppose that the sub... Let f be in the localized nonisotropic Sobolev space Wloc^1,p (H^n) on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n + 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n |△H^n f|^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞(H^n) under the norm of (∫H^n |f| Qp/Q-p)^Q-p/Qp + (∫ H^n |△H^n f|^p)^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L(2Q/Q-2) Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n. 展开更多
关键词 Heisenberg group Sobolev inequalities Poincaré inequalities best constants
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