A necessary condition for local solvability for a class of operators with double characteristics

Local Solvability for a Class of Partial Differential Operators with Double Characteristics

We say that a linear partial differential operator L is locally solvable at the origin if there exists an open neighborhood V of 0 such that for every f ∈ C 0 (V ) there exists u ∈ D(V ) satisfying Lu = f in V . Elliptic (linear) partial differential operators are always locally solvable, as are many others. The example ∂x + i(∂y + x∂t) of Lewy [10] demonstrated that local solvability does not ...

Local solvability of second order differential operators with double characteristics I: Necessary conditions

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic second order differential operators. Consider doubly characteristic differential operators of the form

Local solvability of linear differential operators with double characteristics I: Necessary conditions

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators L, defined, say, in an open set Ω ⊂ Rn. Suppose the principal symbol pk of L vanishes to second order at (x0, ξ0) ∈ T ∗Ω \ 0, and denote by QH the Hessian form associated to pk on T(x0,ξ0)T ∗Ω. As the main result of this paper, we show (under some rank con...

A Necessary Condition for Weyl-Heisenberg Frames

BMO solvability and the A∞ condition for elliptic operators

We establish a connection between the absolute continuity of elliptic measure associated to a second order divergence form operator with bounded measurable coefficients with the solvability of an endpoint BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO Dirichlet problem implies Lp...