On the infinitesimal generator of the Poisson operator II. The n-dimensional case

In a recent paper (5) the author gave an alternative proof of the result due to Hille(2) for the representation of the infinitesimal generator of the Poisson operator. In this paper we obtain the n-dimensional analogue of the result by using arguments similar to those employed in (5).

On operators which depend on some parameter

The purpose of this paper is to study the mapping properties of certain operators which depend on a variable parameter. We prove a general theorem which is applied to some special cases. Among results obtained are new inequalities involving the Poisson operator and its conjugate.

On the infinitesimal generator of the Poisson operator

Throughout this paper we shall denote the space Lp(− ∞, ∞) by Lp, and the numbers p and p′ will be related by 1/p + 1/p′ = 1.The Poisson operator Pa is defined by