Triebel–Lizorkin spaces with non-doubling measures
Volume 162 / 2004
Abstract
Suppose that $\mu $ is a Radon measure on ${{{{\mathbb R}}}^d},$ which may be non-doubling. The only condition assumed on $\mu $ is a growth condition, namely, there is a constant $C_0>0$ such that for all $x\in \mathop {\rm supp}(\mu )$ and $r>0,$ $$\mu (B(x, r))\le C_0r^n,$$ where $0< n\leq d.$ The authors provide a theory of Triebel–Lizorkin spaces ${\dot F^s_{pq}(\mu )}$ for $1< p< \infty $, $1\le 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 does not use the vector-valued maximal function inequality of Fefferman and Stein and 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 given.
