Triebel–Lizorkin spaces with non-doubling measures

Yongsheng Han¹ ; Dachun Yang²

¹ Department of Mathematics Auburn University Auburn, AL 36849-5310, U.S.A.
² Department of Mathematics Beijing Normal University Beijing 100875 People's Republic of China

Studia Mathematica, Tome 162 (2004) no. 2, pp. 105-140

Résumé

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.

Détail

DOI : 10.4064/sm162-2-2

Keywords: suppose radon measure mathbb which may non doubling only condition assumed growth condition namely there constant mathop supp where leq authors provide theory triebel lizorkin spaces dot infty infty theta where theta real number which depends non doubling measure method does vector valued maximal function inequality fefferman stein even classical applications lifting properties these spaces using riesz potential operators dual spaces given