Fundamental solutions for translation and rotation invariant differential operators on the Heisenberg group

Let ${\displaystyle H_1}$ be the three-dimensional Heisenberg group. Consider the left invariant differential operators of the form D=P(-iT,-L), where P is a polynomial in two variables with complex coefficients, L is the sublaplacian on ${\displaystyle H_1}$ and T is the derivative with respect to the central direction. We find a fundamental solution of D, whose definition is related to the way the plane curve defined by P(x,y)=0 intersects the Heisenberg fan F = A ∪ B, A = {(x,y)∈ ℝ^2: y=(2m+1)|x|, m ∈ ℕ, B= {(x,y) ∈ ℝ^2: x=0, y<0}. We can write an explicit expression of such a fundamental solution when the curve P(x,y)=0 intersects F at finitely many points, all belonging to A and, if one of them is the origin, the monomial ${\displaystyle y^k}$ has a nonzero coefficient, where k is the order of zero at the origin. As a consequence, such operators are globally solvable on ${\displaystyle H_1}$.