Subordination by orthogonal martingales in $L^{p}$ and zeros of Laguerre polynomials

TL;DR

This paper determines the optimal $L^p$-norm constants for orthogonal martingale transforms, linking these constants to zeros of Laguerre polynomials, and applies the results to estimate the Ahlfors--Beurling operator's norm.

Contribution

It provides the sharp $L^p$-norm bounds for orthogonal martingale transforms and connects these bounds to Laguerre polynomial zeros, offering new insights and estimates.

Findings

Sharp $L^p$-norm constants for orthogonal martingales when $1<p<2$ and $p>2$.

Explicit connection between martingale bounds and zeros of Laguerre polynomials.

New asymptotic estimate for the Ahlfors--Beurling operator's norm.

Abstract

In this paper we address the question of finding the best $L^p$-norm constant for martingale transforms with one-sided orthogonality. We consider two martingales on a probability space with filtration $\mathcal{B}$ generated by a two-dimensional Brownian motion $B_t$. One is differentially subordinated to the other. Here we find the sharp estimate for subordinate martingales if the subordinated martingale is orthogonal and $1<p<2$, and we find the best constant if $p>2$, but the orthogonal martingale is a subordinator. The answers are given in terms of zeros of Laguerre polynomials. As an application of our sharp constant we obtain a new estimate for the norm of theAhlfors--Beurling operator. We estimate it as $1.3922(p-1)$ asymptotically for large $p$.