LOCAL SOLVABILITY FOR A CLASS OF PARTIAL

DIFFERENTIAL OPERATORS WITH DOUBLE CHARACTERISTICS

MICHAEL CHRIST GEORGI KARADZHOV

Abstract. A necessary and sufficient condition for local solvability is presented for the linear partial differential operators −X^{2} − Y^{2} + ia(x)[X, Y] in R^{3} = {(x, y, t)}, where X = ∂_{x}, Y = ∂_{y} + x^{k}∂_{t}, and a ∈ C^{∞}(R^{1}) is real valued, for each positive integer k.