Lp-Boundedness for the Littlewood-Paley g-Function Connected with the Riemann-Liouville Operator

Abstract

We study the Gauss and Poisson semigroups connected with the Riemann- Liouville operator deflned on the half plane. Next, we establish a principle of maximum for the singular partial difierential operator ¢fi = @ 2 @r2 + 2fi + 1 r @ @r + @ 2 @x2 + @ 2 @t2 ; (r;x;t) 2 )0;+1(£R£)0;+1(: Later, we deflne the Littlewood-Paley g-function and using the principle of maximum, we prove that for every p 2 )1;+1(, there exists a positive constant Cp such that for every f 2 L p (dfi), 1 Cp jjfjjp; fi 6 jjg(f)jjp;fi 6 Cp jjfjjp;fi: