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Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces. (English) Zbl 0806.42013
The concept of space of homogeneous type was introduced by {\it R. R. Coifman} and {\it G. Weiss} as the natural frame for a general Calderón-Zygmund theory [“Analyse harmonique noncommutative sur certains espaces homogènes” (1971; Zbl 0224.43006)]. The authors develop a version of Littlewood-Paley theory in this general context. Their analogue of the Littlewood-Paley decomposition is based on a construction of {\it G. David}, {\it J. L. Journé} and {\it S. Semmes} [Rev. Mat. Iberoam. 1, No. 4, 1-56 (1985; Zbl 0604.42014)]. The authors prove a suitable analogue of the Calderón reproducing formula and apply it to study Besov- and Triebel-Lizorkin spaces defined via the generalized Littlewood-Paley decompositions.

MSC:
42B25Maximal functions, Littlewood-Paley theory
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B30$H^p$-spaces (Fourier analysis)
42B15Multipliers, several variables