On a Class of Calderón-Zygmund Operators Arising from Projections of Martingale Transforms

Abstract

We prove that a large class of operators, which arise as the projections of martingale transforms of stochastic integrals with respect to Brownian motion, as well as other closely related operators, are in fact Calderon-Zygmund operators. These operators have played an important role in studying the Lp boundedness, 1 < p < ∞, of classical Calderon-Zygmund operators such as the Beurling-Ahlfors transform and the Riesz transform. Showing that these operators are Calderon-Zygmund implies that they are not only bounded on Lp, but also satisfy weak-type inequalities. Unlike the boundedness on Lp, which can be obtained directly from the Burkholder martingale transform inequalities, the weak-type estimates do not follow from the corresponding martingale results. The reason for this is that the Lp boundedness of these operators uses conditional expectation, which unfortunately does not preserve weak-type inequalities. Instead, we represent these operators in a purely analytic fashion as integration against a kernel and obtain our result by showing that our kernel satisfies suitable estimates.