On Bessel-Riesz operators

Abstract

This article deals with certain kind of potential operator defined as convolution with the generalized function Wα (P ± i0,m,n)depending on a complex parameter α and a real non negative one m. The definitory formulae and several properties of the family {W (P ± i m n)} α∈C α 0, , α; have been introduced and studied by Trione (see [14]) specially the important followings two: a) Wα ∗Wβ =Wα+β , α and β complex numbers, and b) k W −2 is a fundamental solution of the k-times iterated Klein-Gordon operator Writing Wα (P ± i0,m,n) as an infinite linear combination of the ultrahyperbolic Riesz kernel of different orders Rα (P ± i0)which is a causal (anticausal) elementary solution of the ultrahyperbolic differential operator and taking into account its Fourier transform it is possible to evaluate the Fourier transform of the kernel Wα (P ± i0,m,n). We prove the composition formula Wα ∗Wβϕ =Wα+βϕ for a sufficiently good function. The proof of this result is based on the composition formulae presented by Trione in [14], but we also present a different way. Other simple property studied is the one that establish the relationship between the ultrahyperbolic Klein-Gordon operator and the Wα Bessel-Riesz operator. Finally we obtain an expression that will be consider a fractional power of the Klein-Gordon operator.