The integral represenatation of the riesz kernel W_α(u,m)

Abstract

The Riesz kernel where n is the dimension of the space (cf. form. (1)) can be expressed by means of a Riemann-Liouville integral (cf. form. (13)). The very interesting particular case α=2k expressed by form. (15). The kernel W_α(u,m) has analogous properties to the Riesz kernel (cf. [1]). That is,putting p=1 in (1) we obtain the kernel introduced by Riesz (cf. [1], p.17,[2], p.89, [3], p.179 and [4], p.72). Therefore, we know that ,where

is the n-dimensional ultrahyperbolic Klein-Gordon operator, (cf. [5], p.9, form. (2.29)); (cf. [5], p.19, form (IV.9)), here is the n-dimensional ultrahyperbolic Klein-Gordon operator iterated K-times W₀(u,m)=δ (cf. [5], p.20, form (VI)). OtherwiseW_α(u,m) can be expressed as an infinite, linear combination of R_α(u) of different orders (cf. [5], p.15, form. (3.26)) where R_α(u) is the ultrahyperbolic Riesz (cf. [6], p.72) and W_α(u,m)=R_α(u) (cf. [5], p.22, form. (V2.5). We also study the particular case W₀(u,m) (see Note).

Keywords:

• Riemann-Liouville integral
• ultrahyperbolic Klein-Gordon operator
• integral representation

MSC(1991):

• 45P05

Additional information

Notes on contributors

Susana Elena Trione