Besov spaces with non-doubling measures

Abstract:

Suppose that $\mu$ is a Radon measure on ${\mathbb R}^d,$ which may be non-doubling. The only condition on $\mu$ is the growth condition, namely, there is a constant $C_0>0$ such that for all $x\in {\rm { supp }}(\mu )$ and $r>0,$ \[ \mu (B(x, r))\le C_0r^n,\] where $0<n\leq d.$ In this paper, the authors establish a theory of Besov spaces $\dot B^s_{pq}(\mu )$ for $1\le p, q\le \infty$ and $|s|<\theta$, where $\theta >0$ is a real number which depends on the non-doubling measure $\mu$, $C_0$, $n$ and $d$. The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.