Equations with constant coefficients invariant under a group of linear transformations

Abstract

If P P is a linear differential operator on R n {{\mathbf {R}}^n} with constant coefficients, which is invariant under a group G G of linear transformations, it is not true in general that the equation P u = f Pu = f always has a G G -invariant solution u u for a G G -invariant f f . We elucidate here the particular case of a “big” group G G , and we count the invariant solutions when they exist (see Corollary 28 and Theorems 32, 33). The case, of special interest, of the wave equation and the Lorentz group is covered (Corollary 27). The theory of hyperfunctions provides the frame for the work.