On the weak solutions of the equation related to the Diamond operator (Q2713921)

The authors study the Green function of the operator NEWLINE\[NEWLINE \oplus^k = \Biggl[\biggl(\sum_{i=1}^p\frac{\partial^2}{\partial x_i^2}\biggr)^4 - \biggl(\sum_{j= p + 1}^{p + q}\frac{\partial^2}{\partial x_j^2}\biggr)^4 \Biggr]^k, NEWLINE\]NEWLINE where \(n\) and \(k\) are positive numbers, \(p + q = n\), and \(x = (x_1,\dots, x_n) \in \mathbb C^n\). As a result, the obtained Green function of the operator \(\oplus^k\) is used for constructing a weak solution to the equation NEWLINE\[NEWLINE \oplus^ku(x) = f(x), NEWLINE\]NEWLINE where \(f\) is a given generalized function.