IJPAM: Volume 72, No. 3 (2011)

ELEMENTAL SOLUTIONS OF THE OPERATOR $L^{K}$ AND
THE DISTRIBUTIONAL PRODUCT BETEWEEN
$pf\left\{ P^{-j}\right\}$ AND $\nabla\left( L^{k}P_{+}^{k-\frac{n}{2}}\right)$

Manuel A. Aguirre T.
Núcleo Consolidado Matemática Pura y Aplicada
Facultad de Ciencias Exactas
UNCentro, Pinto 399, 7000m, Tandil, ARGENTINA


Abstract. The object of this paper is obtain elemental solutions of the operator $L^{k}$ and give a sense to distribution products between $pf\left\{ P^{-j}\right\}$ and $\nabla\left( L^{k}P_{+}^{k-\frac{n}{2}}\right)$where $\nabla$ is the operator defined by([*]) and $L^{k}$ is the $n-$dimensional ultrahyperbolic operator iterated $k$-times defined by the formula ([*]). As consequence our formulae([*]) is a genralization of the formula

\begin{displaymath} \Delta^{k}\left\{ r^{2k-n}(A_{k,n}\log r+B_{k,n})\right\} =\delta \end{displaymath}

which appear in (see [#!S!#], p. 47) where $A_{k,n}$ is defined by([*] )and $B_{k,n}$ by ([*]). Our product

\begin{displaymath} pf\left\{ P^{-j}\right\} .\nabla\left( L^{k}P_{+}^{k-\frac{n}{2}}\right) \end{displaymath}

generalizes of the product $r^{-2k}.\nabla(\Delta r^{2-m})$ given by Li Chen Kuan in (see [#!C!#], p. 346).

Received: February 2, 2011

AMS Subject Classification: 46F10, 46F12

Key Words and Phrases: theory of distributions, distributional product

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 72
Issue: 3