Relations between harmonic analysis associated with two differential operators of different orders (Q1874141)

The problem of deducing harmonic analysis associated with a differential operator in the complex domain (generalized translation operators, generalized convolution, generalized Fourier transform and generalized Paley-Wiener theorem) from the one associated with another operator of the same order has been treated in [\textit{J. Delsarte} and \textit{J. L. Lions}, Comment. Math. Helv. 32, 113-128 (1957; Zbl 0080.29501)]. The aim of the present paper is to solve this problem for different operators having different orders. More precisely, a suitable class is considered of differential operators \(L_z\) in the complex domain and from harmonic analysis associated with \(L_z\), the corresponding one associated with \(L^n_z\), \(n\) being an arbitrary positive integer, is stated. Some particular cases are analyzed.