Details of the Previous Chapter

Abstract

In this chapter we provide the details of the previous chapter, some of which may seem rather forbidding. The aim was to prove local real analytic hypoellipticity (at the origin) in both the $\overline{\partial }$-Neumann problem and for □  _b , for pseudoconvex domains that are locally of the form ℑw > h( | z | ²) with h(z) real analytic (and not constant), h(0) = 0. These results were announced in [DT9] and presented in extended form in [DT1]. Again, we use purely L ² methods generalizing those of Tartakoff [T4], [T5], and use results of Derridj and Grigis–Sjöstrand [D3], [GS] that show that one has “maximal” estimates for such domains. (It is worth noting that we make no use of subellipticity per se; a compactness estimate would suffice.) We also need to construct a special holomorphic vector field M that in a sense takes the place of the modifications of ψT cited above and used in [T4].