Quantum differentiability on noncommutative Euclidean spaces

Abstract

We study the topic of quantum differentiability on quantum Euclidean $d$-dimensional spaces (otherwise known as Moyal $d$-spaces), and we find conditions that are necessary and sufficient for the singular values of the quantised differential to have decay $O(n^{-\alpha })$ for $0<\alpha \le \frac{1}{d}$. This result is substantially more difficult than the analogous problems for Euclidean space and for quantum $d$-tori.

Accepted to Comm. Math. Phys