Singular Integrals and Approximate Identities on Spaces of Homogeneous Type1 by Hugo Aimar

In this paper we give conditions for the L2-boundedness of singular integrals and the weak type (1,1) of approximate identities on spaces of homogeneous type. Our main tools are Cotlar's lemma and an extension of a theorem of Z6. Introduction. The behavior of singular integrals and approximate identities as operators on the space of integrable functions, i.e. the weak type (1,1), can be investigated by using the Calderon-Zygmund method. This method relies, essentially, on the possibility of solving two problems of different nature: I. produce an adequate decomposition of If functions, II. prove the Lp boundedness of the operator for some p cz (1, oo ]. Problem I can be solved in the very general setting of spaces of homogeneous type introduced by R. Coifman and M. de Guzman in [CG]. In this paper we study problem II and its application to prove the weak type (1,1) of singular integrals and approximate identities operators with kernels defined on spaces of homogeneous type. The approximate identities considered here are natural generalizations to spaces of homogeneous type of those introduced in [Z]. The main results are the L2 boundedness of singular integrals and the weak type (1,1) of approximate identities. To prove them we impose an additional geometric condition on the normalized homogeneous structure, that is, the boundedness of the measure of an annulus by the difference of its radii. The precise definition is given in §1, where we also include several examples of spaces endowed with this property. The central tool in the proof of L2 boundedness of singular integral operators, given in §3, is Cotlar's lemma. A general class of approximate identities is introduced and studied in §4. We use an extension of the theorem of Z6 (see [Z]) to the general setting of spaces of homogeneous type. In order to obtain this extension we show in §2 a covering lemma and a decomposition lemma for If functions (i.e. we give a solution for problem I) in the case when the space is not necessarily bounded. 1. Definitions and notation. Let X he a set, let a nonnegative symmetric function d on X X X be called a quasi-distance if there exists a constant k such that (1.1) d(x,y)<k[d(x,z) + d(z,y)\ for every x, y, z cz X, and d(x, y) = 0 if and only if x = y. Received by the editors September 28, 1984. 1980 Mathematics Subject Classification. Primary 42B20, 42B25.