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Variable Hardy Spaces Associated with Operators Satisfying Davies–Gaffney Estimates

Part of: Special classes of linear operators Harmonic analysis in several variables Elliptic equations and systems

Published online by Cambridge University Press:  21 May 2018

Dachun Yang ,
Junqiang Zhang  and
Ciqiang Zhuo
Dachun Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China (dcyang@bnu.edu.cn; zhangjunqiang@mail.bnu.edu.cn)
Junqiang Zhang*
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China (dcyang@bnu.edu.cn; zhangjunqiang@mail.bnu.edu.cn)
Ciqiang Zhuo
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, People's Republic of China (cqzhuo@mail.bnu.edu.cn)
*
*Corresponding author.
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Abstract

Let L be a one-to-one operator of type ω in L2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Let p(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space $H_L^{p(\cdot )} ({\open R}^n)$ associated with L. By means of variable tent spaces, the authors establish the molecular characterization of $H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of $H_L^{p(\cdot )} ({\open R}^n)$ is the bounded mean oscillation (BMO)-type space ${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, where L* denotes the adjoint operator of L. In particular, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H_L^{p(\cdot )} ({\open R}^n)$ and show that the fractional integral L−α for α∈(0, (1/2)] is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to $H_L^{q(\cdot )} ({\open R}^n)$ with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇ L−1/2 is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to the variable Hardy space Hp(·)(ℝn).

Keywords

second-order divergence form elliptic operatorDavies–Gaffney estimatevariable Hardy spacesquare functionmaximal functionmoleculeRiesz transform

MSC classification

Primary: 42B30: $H^p$-spaces
Secondary: 42B35: Function spaces arising in harmonic analysis 42B25: Maximal functions, Littlewood-Paley theory 35J15: Second-order elliptic equations 47B06: Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 759 - 810
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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