On the convolution equation related to the diamond kernel of Marcel Riesz

In this paper, we study the distribution e~t~k6 where O* is introduced and named as the Diamond operator iterated k-times (k = 0, 1,2,. . .) and is definded by -~ + N + " " " + 8tz J ~ + z~T+ " " " p+ 2 8~p+q J J ' where t=(h , t2 , . . . , t . ) is a variable and ~=(~1,~2 . . . . ,~.) is a constant and both are the points in the n-dimensional Euclidean space R", $ is the Dirac-delta distribution with 0 ° 6 6 and p + q = n (the dimension of R") At first, the properties of C '0~6 are studied and later we study the application of c t ok 6 for solving the solutions of the convolution equation m * u(t) = e ~' E cr~r(~. r=O We found that its solutions related to the Diamond Kernel of Marcel Riesz and moreover, the type of solutions such as, the classical solution (the ordinary function) or the tempered distributions depending on m, k and ct. (~) 1998 Elsevier Science B.V, All rights reserved. A M S classification: 46F10