Maximal functions for Weinstein operator

In the present paper, we study in the harmonic analysis associated to the Weinstein operator, the boundedness on L^p of the uncentered maximal function. First, we establish estimates for the Weinstein translation of characteristic function of a closed ball with radius ɛ centered at 0 on the upper half space ℝ^d–1× ]0, +∞ [. Second, we prove weak-type L¹-estimates for the uncentered maximal function associated with the Weinstein operator and we obtain the L^p-boundedness of this operator for 1 < p ≤ + ∞. As application, we define a large class of operators such that each operator of this class satisfies these L^p-inequalities. In particular, the maximal operator associated respectively with the Weinstein heat semigroup and the Weinstein-Poisson semigroup belong to this class.