Burkholder inequalities for submartingales, Bessel processes and conformal martingales

TL;DR

This paper establishes optimal inequalities for conformal martingales and submartingales, extending previous results and applying to Bessel processes and Euclidean domain functions, with implications for harmonic analysis.

Contribution

It provides the first sharp bounds for the norms of conformal martingales in arbitrary dimensions, extending to non-integer dimensions and linking to Bessel processes.

Findings

Derived optimal constants for conformal martingale inequalities

Extended inequalities to non-integer dimensions and Bessel processes

Connected results to the behavior of smooth functions on Euclidean domains

Abstract

The motivation for this paper comes from the following question on comparison of norms of conformal martingales $X$, $Y$ in $\R^d$, $d\geq 2$. Suppose that $Y$ is differentially subordinate to $X$. For $0<p<\infty$, what is the optimal value of the constant $C_{p,d}$ in the inequality $$ Y_p\leq C_{p,d}X_p ?$$ We answer this question by considering a more general related problem for nonnegative submartingales. This enables us to study extension of the above inequality to the case when $d>1$ is not an integer, which has further interesting applications to stopped Bessel processes and to the behavior of smooth functions on Euclidean domains. The inequality for conformal martingales, which has its roots on the study of the $L^p$ norms of the Beurling-Ahlfors singular integral operator \cite{BW}, extends a recent result of Borichev, Janakiraman and Volberg \cite{BJV2}.