Calderón--Zygmund Operators Associated to Ultraspherical Expansions

http://dx.doi.org/10.4153/CJM-2007-052-2
Canad. J. Math. 59(2007), 1223-1244

  Published:2007-12-01
 Printed: Dec 2007

We define the higher order Riesz transforms and the Littlewood--Paley $g$-function associated to the differential operator $L_\l f(\theta)=-f''(\theta)-2\l\cot\theta f'(\theta)+\l^2f(\theta)$. We prove that these operators are Calder\'{o}n--Zygmund operators in the homogeneous type space $((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted, $H^1-L^1$ and $L^\infty-BMO$ inequalities are obtained.