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

The transmutation operators were introduced by Delsarte and Lions to state relations between harmonic analysis (generalized translation operators, generalized convolution, generalized Fourier transform and generalized Paley–Wiener theorem) associated with two differential operators of the same order in the complex domain. In this paper, we discuss the analogous problem for differential operators having different orders. More precisely, we consider a suitable class of differential operators Lz in the complex domain and from harmonic analysis associated with Lz, we state the corresponding one associated with Lzn, n being an arbitrary positive integer. Our analysis is based on Ricciʹs decomposition. Some particular cases are singled out.