Algebra properties for Besov spaces on unimodular Lie groups

TL;DR

This paper characterizes Besov spaces on unimodular Lie groups using heat kernel estimates and demonstrates their algebra property when intersected with bounded functions, applicable to groups with various volume growths.

Contribution

It provides new characterizations of Besov spaces on unimodular Lie groups and establishes their algebra property under broad volume growth conditions.

Findings

Besov spaces characterized via heat kernel estimates.

Algebra property established for Besov spaces intersected with L-infinity.

Results applicable to groups with polynomial and exponential volume growth.

Abstract

We consider the Besov space $B^{p,q}_\alpha(G)$ on a unimodular Lie group $G$ equipped with a sublaplacian $\Delta$. Using estimates of the heat kernel associated with $\Delta$, we give several characterizations of Besov spaces, and show an algebra property for $B^{p,q}_\alpha(G) \cap L^\infty(G)$ for $\alpha>0$, $1\leq p\leq+\infty$ and $1\leq q\leq +\infty$. These results hold for polynomial as well as for exponential volume growth of balls.