L^p-Boundedness for the Littlewood-Paley g-Function Connected with the Riemann-Liouville Operator.

RACHDI, LAKHDAR TANNECH; AMRI, BESMA; CHETTAOUI, CHIRINE

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 operator Δα = ∂2/∂r2 2α 1/r ∂/∂r ∂2/∂x2 ∂2/∂t2; (r, x, t) ∊ ]0, ∝[×R×]0, ∝[. Later, 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 Cp such that for every f ∊ Lp(dνα), 1/Cp ‖f‖p;να ≤ ‖g(f)‖p;να ≤ Cp ‖f‖p;να.

LITTLEWOOD-Paley theory; RIEMANNIAN geometry; LIOUVILLE'S theorem; GAUSSIAN function; FOURIER analysis

Kyungpook Mathematical Journal, 2016, Vol 56, Issue 1, p185

1225-6951

Academic Journal

10.5666/KMJ.2016.56.1.185