Sharp Lower bound estimates for vector-valued and matrix-valued multipliers in $L^p$

TL;DR

This paper extends the concept of multipliers to vector and matrix forms, providing sharp lower bound estimates for their $L^p$ operator norms, including a new proof for the Ahlfors-Beurling operator.

Contribution

It introduces a generalized approach to multipliers, deriving sharp lower bounds for vector and matrix-valued cases, and offers a novel proof for the Ahlfors-Beurling operator norm bound.

Findings

Computed sharp lower bounds for quadratic perturbations of the Ahlfors-Beurling operator.

Established a lower bound of p*-1 for the $L^p$ norm of the Ahlfors-Beurling operator.

Generalized multiplier concepts to vector and matrix forms.

Abstract

We generalize the idea of a multiplier in two different ways and generalize a recent result of Geiss, Montomery-Smith and Saksman. First of all, we consider multipliers in the form of a vector acting on a scalar function. Using this technique we compute the sharp lower bound estimate for $L^p$ operator norm of a quadratic perturbation of the real part of the Ahlfors-Beurling operator. Secondly, we consider matrix-valued multipliers to obtain a new proof showing that the $L^p$ operator norm of the Ahlfors-Beurling operator is bounded below by p^*-1.