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

Rachdi, Lakhdar Tannech;Amri, Besma;Chettaoui, Chirine

  • Received : 2015.02.02
  • Accepted : 2015.11.03
  • Published : 2016.03.23

Abstract

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 $${\Delta}_{\alpha}={\frac{{\partial}^2}{{\partial}r^2}+{\frac{2{\alpha}+1}{r}{\frac{\partial}{{\partial}r}}+{\frac{{\partial}^2}{{\partial}x^2}}+{\frac{{\partial}^2}{{\partial}t^2}}};\;(r,x,t){\in}]0,+{\infty}[{\times}{\mathbb{R}}{\times}]0,+{\infty}[$$. Later, we define the Littlewood-Paley g-function and using the principle of maximum, we prove that for every $p{\in}]1,+{\infty}[$, there exists a positive constant $C_p$ such that for every $f{\in}L^p(d{\nu}_{\alpha})$, $${\frac{1}{C_p}}{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}{\leqslant}{\parallel}g(f){\parallel}_{p,{\nu}_{\alpha}}{\leqslant}C_p{\parallel}f{\parallel}_{p,{\nu}_{\alpha}}$$.

Keywords

Riemann-Liouville operator;Fourier transform;Semigroup;Principle of maximum;Littlewood-Paley g-function

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