# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 40 (2011) pp.111-147

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### Abstract

In this paper we introduce a new function space which unifies and generalizes the Besov-type and the Triebel-Lizorkin-type function spaces introduced by S. Jaffard and D. Yang- W. Yuan. This new function space covers the Besov spaces and the Triebel-Lizorkin spaces in the homogeneous case, and further the Morrey spaces. We define the new function space through wavelet expansions. We establish characterizations of the new function space such as the ϕ-transform characterization in the sense of Frazier-Jawerth, the atomic and molecular decomposition characterization. Moreover, in the inhomogeneous case, we give a characterization by local polynomial approximation. As application, we investigate the boundedness of the Calderòn-Zygmund operator and the trace theorem on the new function space.

MSC(Primary) 42B35 42C40, 42B20 wavelet, Besov space, Triebel-Lizorkin space, trace theorem, Calderon-Zygmund operator, atomic and molecular decomposition