Annales Academić Scientiarum Fennicć
Mathematica
Volumen 36, 2011, 509-529

REAL INTERPOLATION FOR GRAND BESOV AND TRIEBEL-LIZORKIN SPACES ON RD-SPACES

Xiaojuan Jiang, Dachun Yang and Wen Yuan

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

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; wyuan 'at' mail.bnu.edu.cn

Abstract. Let X be an RD-space, namely, a metric space enjoying both doubling and reverse doubling properties. In this paper, for all s \in [-1,1] and p,q \in (0,\infty], the authors introduce the grand Besov spaces A\dot{B}^s_p,q(X) and grand Triebel-Lizorkin spaces A\dot{F}^s_p,q(X), and prove that when \epsilon \in (0,1), |s| < \epsilon and p \in (max{n/(n + \epsilon),n/(n + \epsilon + s)},\infty], A\dot{B}^s_p,q(X) \cap (\mathring{G}₀^\epsilon(\beta,\gamma))' = \dot{B}^s_p,q(X) with q \in (0,\infty] and A\dot{F}^s_p,q(X) \cap (\mathring{G}₀^\epsilon(\beta,\gamma))' = \dot{F}^s_p,q(X) with q \in (max{n/(n + \epsilon),n/(n + \epsilon + s)},\infty] for all admissible \beta and \gamma, where \mathring{G}₀^\epsilon(\beta,\gamma) is the space of test functions. As applications, the authors obtain some real interpolation results on these grand Besov and Triebel-Lizorkin spaces. The corresponding results for inhomogeneous spaces are also presented.

2010 Mathematics Subject Classification: Primary 42B35; Secondary 46B70, 46E35, 43A99.

Key words: K-method, Besov space, Triebel-Lizorkin space, Calderón reproducing formula.

Reference to this article: X. Jiang, D. Yang and W. Yuan: Real interpolation for grand Besov and Triebel-Lizorkin spaces on RD-spaces. Ann. Acad. Sci. Fenn. Math. 36 (2011), 509-529.

Full document as PDF file

doi:10.5186/aasfm.2011.3635