Symbol calculus of pseudo-differential operators on Spin(4)

During the last decade, a new and full symbol calculus over compact groups was developed by M. Ruzhansky, V. Turunen, and J. Wirth which represents a non-commutative extension of the classical Kohn-Nirenberg quantization. This calculus has several advantages over the classic principle calculus of L. Hormander, which is based on the notion of the symbol via localizations, such as the characterization of global and local hypoellipticity.

In this seminar, we present a full symbol calculus of pseudo-differential operators on the group Spin(4). The essential tools for such calculus are the Spin(4)-representations, its matrix coefficients, recurrence relations, difference operators acting on them, and the Fourier transform on Spin(4). Spin(4)-representations are constructed in the spaces of simplicial harmonic and spinor-valued monogenic polynomials using tools from Clifford analysis. Since Spin(4) is isomorphic to the direct product group of Spin(3) with itself, Spin(4)-representations decompose as the tensor product of Spin(3)-representations. With all the tools in hand, we characterize elliptic and global hypoelliptic pseudo-differential operators in Spin(4), in terms of their matrix-valued full symbols. Some examples of first and second-order globally hypoelliptic differential operators will be shown, in particular, of operators that are locally not invertible nor hypoelliptic but globally are.