# On p-elliptic divergence form operators and holomorphic semigroups

## TL;DR

This paper investigates p-elliptic divergence form operators, demonstrating that p-ellipticity ensures the extension of holomorphic semigroups on Lebesgue spaces, with implications for harmonic analysis tools like Hoo-calculi and Riesz transforms.

## Contribution

It establishes that p-ellipticity conditions lead to holomorphic semigroup extrapolation on Lebesgue spaces, extending previous maximal ranges for elliptic operators.

## Key findings

- Holomorphic semigroups extend to a p-dependent range of exponents.
- p-ellipticity implies boundedness of Riesz transforms.
- Results improve understanding of harmonic analysis for divergence form operators.

## Abstract

Second order divergence form operators are studied on an open set with various boundary conditions. It is shown that the p-ellipticity condition of Carbonaro-Dragicevic and Dindos-Pipher implies extrapolation to a holomorphic semigroup on Lebesgue spaces in a p-dependent range of exponents that extends the maximal range for general strictly elliptic coefficients. Results have immediate consequences for the harmonic analysis of such operators, including Hoo-calculi and Riesz transforms.