Global hypoellipticity and global solvability for vector fields on compact Lie groups

Regularity of solutions to a Vekua-type equation on compact Lie groups

“…In [10], M. Ruzhansky, V. Turunen, and J. Wirth have given sufficient conditions for the global hypoellipticity for a class of pseudo-differential operators on compact Lie groups in terms of their matrixvalued full symbols, that will be defined on (1.6). This approach works well in the case where p = 0 (see [5]) because in this case, the operator P is linear. When p = 0, even P being a left-invariant operator, its symbol depends on x ∈ G, so we do not have the expected property Pu(ξ ) = σ P (ξ ) u(ξ ), for every…”

“…where X is a normalized vector field on G and p, q ∈ C, with p = 0. The case p = 0 was studied in [5]. Notice that when p = 0 the operator P is R-linear but is not C-linear.…”

“…for all 1 ≤ m ≤ d ξ . This property implies that the vector field X is globally hypoelliptic (Theorem 3.3 of [5]). Therefore G must be isomorphic to a torus, because the Greenfield-Wallach conjecture is valid for compact Lie groups, proved by Greenfield and Wallach in [3].…”

“…where p, q ∈ C, with p = 0. The case p = 0 was studied in [5]. For u ∈ D ′ (G), we define its conjugatē…”

Regularity of solutions to a Vekua-type equation on compact Lie groups

“…where X is a normalized vector field on G and p, q ∈ C, with p = 0. The case p = 0 was studied in [5]. We say that P is globally hypoelliptic if the conditions u ∈ D ′ (G) and…”