Singular integrals in uniformly convex spaces

Abstract

We consider the action of finitely truncated singular integral operators on functions taking values in a Banach space. Such operators are bounded for any Banach space, but we show a quantitative improvement over the trivial bound in any space with an equivalent uniformly convex norm. This answers a question asked by Naor and the author, who previously proved the result in the important special case of the finite Hilbert transforms. The proof, which splits the operator into a cancellative part and two paraproducts, follows the broad outline of similar results for genuinely singular (non-truncated) operators in the narrower class of UMD spaces. Thus we revisit and survey the recent techniques behind such results, but our precise setting, the main theorem, and some aspects of its proof, are new. Curiously, it turns out that the paraproducts admit somewhat better bounds than the full operator. In a large class of spaces other than UMD, they remain bounded even without the finite truncations.