Behavior of the Bergman projection on the Diederich-Fornæss worm

Abstract

w 1. Introduction In this paper we show that the Bergman projection operator for certain smooth bounded pseudoconvex domains does not preserve smoothness as measured by Sobolev norms. Let Q be a smooth bounded pseudoconvex in C and let p~t) denote the orthogonal projection from L2(f~) onto the Bergman subspace B(ff~)=L2(~) f3 r with respect to the weighted norm I{fll~ J'olf(z)l 2 e-t{lzll2dV)l/2. Let wk(t)) denote the Sobolev space consisting of functions whose derivatives of order ~ -to(k, g2). On the other hand, there is a large collection of results implying that for certain types of domains the unweighted Bergman projection p=pr preserves W k for all k~0. (See [FK] for the strictly pseudoconvex case; for results on weakly pseudoconvex domains the reader may consult [Ko2], [Ca], [Si] and the recent [BStl], [Ch] as well as the references cited therein. Most of these results are focused on the a-Neumann operator rather than P; see [BSt2] for the connection. Also, except for [Kol], positive results in this area are typically valid for any choice of smooth positive weight function on 0.) The question of whether or not P is similarly well-behaved for all weakly pseudoconvex domains has remained open for many years. In this paper we show that this is not the case; in fact when Q is the so-called domain of Diederich and Forna~ss then P does not map W* to W k when k>~zr/(total amount of winding). This latter quantity is explained in section 4 below, where the construction of the worm is reviewed and the main result is proved. The proof depends on computations for a piecewise Levi-flat model depending in turn on certain one-dimensional computations; these are treated in sections 3 and 2, respectively. Section 5 contains additional remarks and questions.