Sharp $L^p$ Convergence for Mirror-Degenerate Expansions

TL;DR

This paper investigates weighted $L^p$ convergence of a reconstruction operator in a non-symmetric harmonic analysis setting, revealing structural properties and conditions for boundedness related to mirror localization.

Contribution

It provides a detailed analysis of the operator's structure and establishes new boundedness criteria based on mirror-local integrability conditions.

Findings

Boundedness characterized by mirror-local integrability of the weight.

Operator admits a structural decomposition after localization.

Results extend understanding of $L^p$ convergence in non-symmetric settings.

Abstract

We analyze weighted $L^p$ convergence for the truncated reconstruction operator in the rank-one non-symmetric Heckman--Opdam setting. After localization at the mirror, the operator admits a rigid structural decomposition and reduces, up to bounded terms, to a rank-one functional. Boundedness on $L^p(w)$ is characterized by the mirror-local integrability of $w^{-\frac{1}{p-1}}$.