# Singular integrals in quantum Euclidean spaces

## TL;DR

This paper develops a noncommutative singular integral and pseudodifferential calculus framework for quantum Euclidean spaces and tori, extending classical theories to noncommutative geometries with new Calderón-Zygmund techniques.

## Contribution

It introduces a novel Calderón-Zygmund theory incorporating nonconvolution kernels for quantum spaces, extending pseudodifferential calculus beyond previous results.

## Key findings

- Established $L_p$-boundedness and Sobolev estimates for various symbols.
- Generalized Calderón-Vaillancourt and Bourdaud theorems to quantum settings.
- Proved $L_p$-regularity of solutions to elliptic PDEs in quantum Euclidean spaces.

## Abstract

In this paper, we establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes' pseudodifferential calculus for rotation algebras, thanks to a new form of Calder\'on-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce $L_p$-boundedness and Sobolev $p$-estimates for regular, exotic and forbidden symbols in the expected ranks. In the $L_2$ level both Calder\'on-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove $L_p$-regularity of solutions for elliptic PDEs.