CHAPTE R 1 Main results Introduction. x The historical development of Calderon-Zymund theory, especially as it applies to end-point spaces and various geometrical settings, has proceeded along two main fronts. The first was concerned with understanding oscillation properties needed to establish basic boundedness results, the T(l)-theorem being perhaps the most fundamental example, while the Pt-Qt methods of Coifman and Meyer replaced sub- tle oscillations restrictions with more effective Carleson measure conditions. The second front dealt with basic building blocks for function spaces and operators, including the representation of Calderon-Zygmund operators as well as their action on the building blocks themselves ([MeYb, FJWa]). For instance, the Calderon Reproducing formula provided a continuous decomposition of the identity operator atoms, molecules and wavelets, on the other hand, were used to provide correspond- ing discrete decompositions, allowing questions of boundedness to be rephrased in terms of more transparent matrix boundedness. Underlying all these developments, either explicitly or implicitly, was the action of the affine group. The introduction of wavelets did not lead, however, to a full understanding of the relationship between the distributional kernels of operators and their represen- tations via various families of molecular building blocks. This occurred for several reasons. Being a basis is too strong a condition to impose on a family of functions - a more flexible, intrinsic restriction is needed. Secondly, the Lipschitz condition traditionally imposed on the kernel of a Calderon-Zygmund operator is not precise enough to guarantee that the operator preserves fully the fundamental properties of such families of building blocks - there is an inevitable loss of smoothness in general. In this memoir we shall address both issues, developing at the same time a theory of smooth decompositions of singular integral operators which unifies all the previous approaches. It employs techniques that are independent of L 2 -theory and of Fourier transforms. The requisite flexibility and Lipschitz smoothness are achieved through the use of frame decompositions with basic building blocks drawn from a Banach space Ms of molecules whose decay, smoothness and cancellation depend on a parameter 5, 0 S oo, and by imposing double Lipschitz bounds on the kernel of Calderon-Zygmund op- erators formulated in terms of a so-called lis-condition. At the expense possibly of some repetition the main results of the memoir will be described in this chapter within the context of known results. Full statements of definitions, results and detailed proofs then appear in the succeeding chapters. deceived by the editor on Oct. 22, 1997. Revised manuscript received on Sep. 12, 2000 1
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