ACTA UNIVERSITATIS APULENSIS No 20/2009 BOUNDEDNESS OF MULTILINEAR COMMUTATOR OF SINGULAR INTEGRAL IN MORREY SPACES ON HOMOGENEOUS SPACES
نویسندگان
چکیده
In this paper, we prove the boundedness of the multilinear commutator related to the singular integral operator in Morrey and Morrey-Herz spaces on homogeneous spaces. 2000 Mathematics Subject Classification: 42B20, 42B25. 1. Preliminaries Sawano and Tanka(see [13]) introduced the Morrey spaces on the non-homogeneous spaces and proved the boundedness of Hardy-Littlewood maximal operators, CalderónZygmund operators and fractional integral operators in Morrey spaces. On the base of the above results, Yang and Meng(see [14]) considered the boundedness of the commutators generated by Calderón-Zygmund operators or fractional integral operators with RBMO(μ) in Morrey spaces. Motivated by these results, in this paper, we will introduce the multilinear commutator related to the singular operator on homogeneous spaces, and prove the boundedness properties of the operator in Morrey and Morrey-Herz spaces on homogeneous spaces. Give a set X, a function d : X ×X → R+ is called a quasi-distance on X if the following conditions are satisfied: (i) for every x and y in X, d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y, (ii) for every x and y in X, d(x, y) = d(y, x), (iii) there exists a constant l ≥ 1 such that d(x, y) ≤ l(d(x, z) + d(z, y)) (1) for every x, y and z in X. Let μ be a positive measure on the σ-algebra of subsets of X which contains the r-balls Br(x) = {y : d(x, y) < r}. We assume that μ satisfies a doubling condition, that is, there exists a constant A such that
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