Convexity and Condition (.Ψ)

Abstract

Most of this chapter is devoted to local existence theory for analytic solutions of the differential equation ∂u/∂z₁ = f near a boundary point of a strictly pseudo-convex domain. This may seem like a very special problem, but in fact, via the transformation theory of microlocal analysis, it is equivalent to the central question of local solvability for microfunctions. This transformation theory is beyond the scope of these lectures, but we shall discuss the geometrical aspects in detail. The whole chapter is an exposition of the thesis of Trépreau [1], but the arguments have been rearranged in order to postpone the reference to microlocal analysis until the very end.