Hardy–Littlewood–Sobolev type inequalities associated with the Weinstein operator

Abstract

We introduce the Riesz potential associated with the Weinstein operator of order s, as where is its heat semigroup. Next, we define the fractional Littlewood–Paley g-function denoted of order s, as where is the Poisson integral of f. We establish, for the two operators, the following Hardy–Littlewood–Sobolev type inequalities. For , p>1 and , there exists a positive constant such that for every in the appropriate Lebesgue space