Minimal Regularity Conditions for the End-point Estimate of Bilinear Calderón-zygmund Operators

A crucial property addressed in the linear Calderón-Zygmund theory, going back to the founding article [1], is the fact that operators bounded on L whose kernels possess certain regularity are in fact bounded on every L space for 1 < p < ∞. Moreover, such a regularity assumption implies, together with the L-boundedness of the operator, a weak-type end-point estimate in L. From these continuity properties, the whole range of values of p follows by duality and interpolation. The quest for the minimal amount of regularity needed to guarantee the existence of such an end-point estimate has a rich history, with several important results that promoted as a byproduct numerous developments in harmonic analysis. For classical singular integrals operators with homogeneous kernels, the question was finally settled in the work of Seeger [23], who showed that the kernel of the operators could be quite rough. See also previous work of Christ [5] and Christ and Rubio de Francia [7]. We refer to [23] and its featured review by Hofmann [13] for precise technical details, an account of the history, and relevant references. For more general multiplier operators, as well as operators of non-convolution type, the regularity of the kernel is very closely related to a Lipschitz-type one. It is convenient for our purposes to recall some well-known facts related to this. Assume that T : L(R) → L(R) is an operator that, at least for x / ∈ supp f , is given by