Local solvability of partial differential operators on the Heisenberg group

Local solvability on $${\mathbb{H}_1}$$ : non-homogeneous operators

“…We start with the more novel techniques beyond those of [14][15][16]. In those articles, as in this one, singularities in the parametrices are a concern as the Fourier parameter η tends to 0.…”

“…Here ζ satisfies W ( ζ ) = Ce γ t 2 /2 for some constant C (cf. [2,14]) where γ def = n j=1 γ j . Therefore, using Abel's formula along with the above estimate, W ( ψ)(t, μ) e γ t 2 /2+a n t/μ for sufficiently large μ 0 where a n = [E 1 ] n,n .…”

“…We choose the realization of the corresponding Lie algebra using X def = ∂ x , Y def = ∂ y + x∂ w for (group) variables x, y, w. Note that our operators include all generators of h C 1 since ∂ w = [X, Y ]. Local solvability for such a class of operators is well researched and we defer to [8,9,14] and the references therein for an introduction to the present research. Our work closely follows those techniques used in [2,[14][15][16] to study related operators.…”

“…The latter result has been applied to various left-invariant operators on H m of various dimensions m [10]. Our operators, however, are at least doubly characteristic (see [14]). In contrast, our approach is, for the most part, to study solvability of operators P(X, Y ) as compared to the solvability of the operator P n (X, Y ), formed by highest-order terms of P(X, Y ) in the subalgebra of h C 1 generated by X and Y -not necessarily those of highest order in differentiation: Note, for instance, that ∂ w is of order two in the subalgebra, but has a symbol of order one.…”

“…From [14] we have immediately Results on non-solvability we are ready to state are as follows: Theorem 1.8 Suppose L is as in (1.1) for some generic P n . Then L is not locally solvable if, for some choice of ± sign, the set of parameters μ ∈ R + : ker(L ± μ ) * S(R)\{0} = ∅ has a limit point in R + .…”