Introduction. Our purpose is to sketch a new approach to proving the boundedness of a vast class of linear operators which includes, e.g., the generalized Calderón-Zygmund operators discussed in [M]. The aproach is based on estimates of operator norms which come from applying recent results concerning the Lp-boundedness of martingale transforms.

In fact, the incentive for this work was the desire to extend some previously known boundedness results for operators acting in Lp-spaces of scalar-valued functions to the case of analogous spaces of X-valued Bochner measurable functions, where X is a Banach space. The recent results, due mainly to D. Burkholder and J. Bourgain, indicated that the class of the so-called UMD-spaces may be exactly the domain in which all results concerning Calderón–Zygmund integral operators and their generalizations remain valid. (Many Banach spaces which are important in classical analysis belong to that class.) Even the simplest singular integral operator, i.e., the Hilbert transform on the real line R, has the property that its natural extension to an operator acting on Lp (RX) where 1 < p < ∞ is a bounded linear map if and only if the Banach space X is a UMD-spa.ce (cf. [Bu2] and [Bo]).

In order to obtain this extension it was necessary to find such proofs which make no use of any result that does not hold in the UMD-setting (for instance, the Fourier transform should be avoided, because it is not bounded in Lp(R, X), unless X is isomorphic to a Hilbert space).