A remark on extrapolation of rearrangement operators on dyadic $H^s$, $0< s \le 1$
Volume 171 / 2005
Studia Mathematica 171 (2005), 196-205
MSC: 46B42, 46B70, 47B37.
DOI: 10.4064/sm171-2-5
Abstract
For an injective map $ \tau $ acting on the dyadic subintervals of the unit interval $[0,1)$ we define the rearrangement operator $ T_s $, $0< s< 2$, to be the linear extension of the map $$ \frac{h_I}{|I|^{1/s}} \mapsto \frac{h_{\tau(I)}}{|\tau(I)|^{1/s}}, $$ where $h_I$ denotes the $L^\infty$-normalized Haar function supported on the dyadic interval $I. $ We prove the following extrapolation result: If there exists at least one $0< s_0< 2$ such that $T_{s_0}$ is bounded on $H^{s_0}$, then for all $0< s< 2 $ the operator $T_{s}$ is bounded on $H^{s}.$
