On the local solvability of a class of degenerate second order operators with complex coefficients

Abstract

We study the local solvability of a class of operators with multiple characteristics. The class considered here complements and extends the one studied in [9], in that in this paper we consider some cases of operators with complex coefficients that were not present in [9]. The class of operators considered here ideally encompasses classes of degenerate parabolic and Schrödinger type operators. We will give local solvability theorems. In general, one has L 2 local solvability, but also cases of local solvability with better Sobolev regularity will be presented.

Sufficient conditions for local solvability of some degenerate pdo with complex subprincipal symbol

“…Note that in this example the vector fields X 1 and X 2 do not form an involutive distribution. This shows that the class (1) generalizes the class of mixed type operators in [6] where the presence of functions f j = f , for all 1 ≤ j ≤ N , is required and a strict sign condition of the form iX 0 (x, D)f > 0 on f −1 (0) is needed. Moreover, with respect to the class of Shrödinger type operators in [6] where, instead, the presence of several functions f j is allowed and X 0 , X N +1 are assumed to be such that X 0 ≡ 0 and X N +1 ≡ 0, here we do not require any involutive structure of the vector fields X j , 1 ≤ j ≤ N , whereas in [6] an involutivity property is considered.…”

“…Some generalizations of this operator have been studied by Colombini, Cordaro and Pernazza in [1] and by Colombini, Pernazza and Treves in [2], and further extensions have been given by the author and A. Parmeggiani in [5] and [6] (see also [22]). The form of the operator P in (1) is mostly linked to that in [5] and [6] and our aim is to cover other open cases. With respect to the class in [6], here we allow the presence of several functions in the second order part of the operator even when X N +1 (x, D) ≡ 0.…”

Local Solvability of Some Partial Differential Operators with Non-smooth Coefficients

“…The local solvability results for the two classes P 1 and P 2 are given at the points of degeneracy where the function appearing in the second order part vanishes. If we consider the classes outside the set S we get operators of the form considered in [8] where a local solvability result outside of S is given by using Carleman estimates. Therefore solvability results out of S are available for these operators.…”

“…The classes P 1 and P 2 above, fully analyzed in [4] (see also [5]), are inspired by previous classes analyzed by Colombini et al in [1] and by Federico and Parmeggiani in [7,8] (see also [24] for a survey), and which are in turn an elaboration of the Kannai operator (see [14]).…”

“…In fact, the hypoellipticity of the Kannai operator on R n is not enough to guarantee the local solvability at the points around which the principal symbol changes sign. However, as shown in [1,7,8], the behaviour of the subprincipal symbol dictates, somehow, the local solvability at the multiple characteristics points.…”