Some fundamental solutions which generalize Fresnel integrals (Q1916803)

The goal of the present paper is to find a fundamental solution for the family of operators \(P= \partial/\partial t+\alpha\partial^m/\partial z^m\), \(m\geq 2\), where \(\partial/\partial z\) is the operator of complex differentiation, \(\partial/\partial z= 1/2(\partial_x-i\partial_y)\). In some sense \(P\) generalizes the heat operator, but this is only formal, because \(\partial/\partial t+\alpha\partial^m/\partial z^m\) is neither hypoelliptic nor parabolic. As usually, the application of partial Fourier transform to \(PE=\delta\) yields an ordinary differential equation depending on a complex parameter. Its solution and the inverse transform motivate to use Fresnel integrals for the description of the fundamental solution. Its singular support is determined at the end of the paper.