# Lusin area integrals related to Jacobi expansions

## TL;DR

This paper studies Lusin area integrals linked to Jacobi polynomial expansions, demonstrating they are Calderón-Zygmund operators and establishing their boundedness properties on weighted spaces.

## Contribution

It shows that mixed Lusin area integrals for Jacobi expansions are vector-valued Calderón-Zygmund operators, enabling the derivation of their mapping properties.

## Key findings

- Operators are Calderón-Zygmund in the Jacobi setting
- Boundedness on weighted L^p spaces established
- Unified approach via space of homogeneous type

## Abstract

We investigate mixed Lusin area integrals associated with Jacobi trigonometric polynomial expansions. We prove that these operators can be viewed as vector-valued Calder\'on-Zygmund operators in the sense of the associated space of homogeneous type. Consequently, their various mapping properties, in particular on weighted $L^p$ spaces, follow from the general theory.