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

We study the Gauss and Poisson semigroups connected with the Riemann-Liouville operator defined on the half plane.Next, we establish a principle of maximum for the singular partial differential operatorLater, we define the Littlewood-Paley g-function and using the principle of maximum, we prove that for every p ∈ ]1, +∞[, there exists a positive constant C p such that for every f ∈ L p (dνα), 1 C p ||f || p,να ||g(f )|| p,να C p ||f || p,να .