A remark on extrapolation of rearrangement operators on dyadic $H^s$, $0 s \le 1$

Stefan Geiss ¹   ; Paul F. X. Müller ²   ; Veronika Pillwein ²

¹ Department of Mathematics and Statistics P.O. Box 35 (MaD) FIN-40014 University of Jyväskylä, Finland
² Department of Analysis J. Kepler University A-4040 Linz, Austria

Studia Mathematica, Tome 171 (2005) no. 2, pp. 196-205

Résumé

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 $H^{s}.$

Détail

DOI : 10.4064/sm171-2-5

Keywords: injective map tau acting dyadic subintervals unit interval define rearrangement operator linear extension map frac mapsto frac tau tau where denotes infty normalized haar function supported dyadic interval prove following extrapolation result there exists least bounded operator bounded nbsp