Cotlar martingale transforms and related singular integrals

Abstract

The "magical" identity discovered by M.~Cotlar in 1955 for the Hilbert transform is established here in the setting of martingale transforms and, in particular, for conformal martingales. This, together with the probabilistic representation of the Riesz transforms, shows that, at the level of martingale transforms and in odd dimensions, they exhibit the same analytic-type structure as the Hilbert transform on the real line. Consequently, Cotlar's proof of the sharp $L^p$ inequality for powers of $2$ applies. The significance of the martingale Cotlar identity, whose proof is entirely elementary, does not lie in providing an alternative proof of this well-known and relatively simple estimate, but rather in the structural viewpoint it reveals. This structure is explored further. Independent of Cotlar's identity, asymptotic bounds for the $L^p$ norm of the vector of Riesz transforms are investigated. It is shown that, in the limit as $p\to\infty$, this norm coincides asymptotically with that of the Hilbert transform on the real line. The study of the Cotlar identity in the martingale setting is motivated by the desire to gain new insight into two longstanding open problems: T.~Iwaniec's 1983 conjecture on the norm of the Beurling-Ahlfors operator and the problem of determining the sharp constant in E.~M.~Stein's 1984 inequality for the vector of Riesz transforms. Related problems are also discussed. The paper contains both a survey of known results and new contributions. An effort has been made to keep the exposition as self-contained as possible and to present the material in an accessible, largely expository style.