IJPAM: Volume 58, No. 4 (2010)

RELATIONS BETWEEN THE ULTRAHYPERBOLIC
OPERATOR AND $(k-1)$-TH DERIVATIVE OF
DIRAC'S DELTA IN $P(x)$ AND $P(x)-c^{2}$

Manuel A. Aguirre
Núcleo Consolidado Matemática Pura y Aplicada (NuCOMPA)
Facultad de Ciencias Exactas
Universidad Nacional del Centro
Pinto 399, Tandil, Provincia de Buenos Aires, 7000, ARGENTINA
e-mail: [email protected]


Abstract.In this article we obtain formulae between the operator ultrahyperbolic iterated $j-$times defined by ([*]) and the $(k-1)$-th derivative of Dirac's delta defined in $P(x)=x_{1}^{2}+\dots x_{\mu }^{2}-x_{\mu +1}^{2}\dots -x_{\mu
+\nu }^{2}$ and $P(x)-c^{2}. $ In particular we obtain that $\delta
^{(k-1)}(P_{+})$ is homogeneous solution of the ultrahyperbolic operator iterated $l$ times if $\frac{n}{2}-l\leq k<\frac{n}{2}$ and $E_{n,r,\mu ,\nu}$ defined in (23) is elementary solution of ultrahyperbolic operator iterated $l$ times. By putting $k=s+1, l=1$ in (18) we have that $\delta
^{(s)}(P_{+})$ is homogeneous solution of the ultrahyperbolic operator if $s=%
\frac{n-4}{2}. $

Received: December 4, 2009

AMS Subject Classification: 46F10, 46F12

Key Words and Phrases: theory of distributions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 58
Issue: 4