The L^p-norms of the Beurling-Ahlfors transform on radial functions

TL;DR

This paper computes the L^p-norms of the Beurling-Ahlfors transform when applied to radial functions, extending previous results to the case p>2 and providing a more complete understanding of its behavior across different p-values.

Contribution

It determines the exact norms of the Beurling-Ahlfors transform on radial functions for p>2, expanding the known range of p for which these norms are understood.

Findings

Calculated the operator norms for p>2

Extended previous results to a broader p-range

Provided explicit norm formulas for radial functions

Abstract

We calculate the norms of the operators connected to the action of the Beurling-Ahlfors transform on radial function subspaces introduced by Ba\~nuelos and Janakiraman. In particular, we find the norm of the Beurling-Ahlfors transform acting on radial functions for $p>2$, extending the results obtained by Ba\~nuelos and Janakiraman, Ba\~nuelos and Os\c{e}kowski, and Volberg for $1<p\leq 2$.