Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 47-87

VARIABLE HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING DAVIES–GAFFNEY ESTIMATES ON METRIC MEASURE SPACES OF HOMOGENEOUS TYPE

Dachun Yang and Junqiang Zhang

Beijing Normal University, School of Mathematical Sciences
Laboratory of Mathematics and Complex Systems, Ministry of Education
Beijing 100875, P.R. China; dcyang 'at' bnu.edu.cn

Beijing Normal University, School of Mathematical Sciences
Laboratory of Mathematics and Complex Systems, Ministry of Education
Beijing 100875, P.R. China; zhangjunqiang 'at' mail.bnu.edu.cn

Abstract. Let (X,d,μ) be a metric measure space of homogeneous type and p(⋅) : X → (0,1] a variable exponent function satisfying the globally log-Hölder continuous condition. Assume that L is a one-to-one operator of type ω on L²(X), with ω ∈ [0,π/2), which has a bounded holomorphic functional calculus, and whose heat kernel satisfies the Davies–Gaffney estimates. In this article, the authors introduce the variable Hardy space H_L^p(⋅)(X) associated with L. Then the authors establish the molecular characterization of H_L^p(⋅)(X) via the atomic decomposition of variable tent spaces and show that the dual space of H_L^p(⋅)(X) is the BMO-type space BMO_p(⋅),L*(X), where L^* denotes the adjoint operator of L on L²(X). In particular, if L is a non-negative self-adjoint operator whose heat kernel has a Gaussian upper bound, the authors then obtain the non-tangential and the radial maximal function characterizations of H_L^p(⋅)(X) via establishing its atomic characterization.

2010 Mathematics Subject Classification: Primary 42B30; Secondary 42B35, 42B25, 47A60, 30L99.

Key words: Metric measure space of homogeneous type, variable Hardy space, Davies–Gaffney estimate, non-negative self-adjoint operator, square function, maximal function, molecule, atom.

Reference to this article: D. Yang and J. Zhang: Variable Hardy spaces associated with operators satisfying Davies–Gaffney estimates on metric measure spaces of homogeneous type. Ann. Acad. Sci. Fenn. Math. 43 (2018), 47-87.

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https://doi.org/10.5186/aasfm.2018.4304