On the norm of the operator $aI+bH$ on $L^p(\mathbb R)$

TL;DR

This paper offers a direct proof for the exact operator norm of linear combinations of the identity and Hilbert transform on real line $L^p$ spaces, avoiding circle-based methods and introducing new extremals for certain cases.

Contribution

It provides a direct proof of the operator norm formula for $aI+bH$ on $L^p(R)$, simplifying previous approaches and introducing new approximate extremals for $p>2$.

Findings

Explicit formula for the norm of $aI+bH$ on $L^p(R)$.

A direct proof avoiding circle-based conjugate function results.

New approximate extremals for the case $p>2$.

Abstract

We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky \cite{HKV}: let $H$ be the Hilbert transform and let $a,b$ be real constants. Then for $1<p<\infty$ the norm of the operator $aI+bH$ from $L^p(\mathbb R)$ to $L^p(\mathbb R)$ is equal to $$ \bigg(\max_{x\in \Bbb R}\frac{|ax-b+(bx+a)\tan \frac{\pi}{2p}|^p+|ax-b-(bx+a)\tan \frac{\pi}{2p}|^p}{|x+\tan \frac{\pi}{2p}|^p+|x-\tan \frac{\pi}{2p}|^p} \bigg)^{\frac 1p}. $$ Our proof avoids passing through the analogous result for the conjugate function on the circle, as in \cite{HKV}, and is given directly on the line. We also provide new approximate extremals for $aI+bH$ in the case $p>2$.