We construct higher-dimensional versions of the Diederich-Fornaess worm domains and show that the Bergman projection operators for these domains are not bounded on high-order L p -Sobolev spaces for 1 ≤ p < ∞.
The Lorenz attractor is one of the best known examples of applied mathematics. However, much of what is known about it is a result of numerical calculations and not of mathematical analysis. As a step toward mathematical analysis, we allow the time variable in the three dimensional Lorenz system to be complex, hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane. Knowledge of singularities being fundamental to any investigation in the complex plane, we build upon earlier work and give a complete and consistent formal development of complex singularities of the Lorenz system using psi series. The psi series contain two undetermined constants. In addition, the location of the singularity is undetermined as a consequence of the autonomous nature of the Lorenz system. We prove that the psi series converge, using a technique that is simpler and more powerful than that of Hille, thus implying a two-parameter family of singular solutions of the Lorenz system. We pose three questions, answers to which may bring us closer to understanding the connection of complex singularities to Lorenz dynamics.
Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let β be a function smooth up to the boundary on a smooth bounded pseudoconvex domain Ω ⊂ C n . We show that, if Ω is convex or the Levi form of the boundary of Ω is of rank at least n − 2, then compactness of the Hankel operator H β implies that β is holomorphic "along" analytic discs in the boundary. Furthermore, when Ω is convex in C 2 we show that the condition on β is necessary and sufficient for compactness of H β .
We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain in C n is an obstruction to compactness of the ∂-Neumann operator on , provided that at some point of M, the Levi form of b has the maximal possible rank n−1−dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. (2000): 32W05 Mathematics Subject Classification
Let Ω be a bounded convex Reinhardt domain in C 2 and φ ∈ C(Ω). We show that the Hankel operator H φ is compact if and only if φ is holomorphic along every non-trivial analytic disc in the boundary of Ω.Let Ω be a domain in C n and let L 2 (Ω) and A 2 (Ω) denote square integrable functions on Ω and the Bergman space on Ω (the set of square integrable holomorphic functions on Ω), respectively. Since A 2 (Ω) is a closed subspace in L 2 (Ω) the Bergman projection P : L 2 (Ω) → A 2 (Ω), the orthogonal projection, exists. Furthermore, let H φ f = (I − P)(φ f ) for all f ∈ A 2 (Ω) and φ ∈ L ∞ (Ω). We note that H φ is called the Hankel operator with symbol φ. We refer the reader to [Pel03, Zhu07] and references there in for more information on these operators.Hankel operators form an active research area in operator theory. Our interest lies in their compactness properties in relation to the behavior of the symbols on the boundary of the domain. On the unit disc D in C Axler ([Axl86]) showed that, for f holomorphic on the unit disc D, the Hankel operator H f is compact on A 2 (D) if and only if f is in the little Bloch space (that is, (1 − |z| 2 )| f (z)| → 0 as |z| → 1). This result has been extended into higher dimensions by Peloso ([Pel94]) in case the domain is smooth bounded and strongly pseudoconvex. The same year, Li ([Li94]) characterized bounded and compact Hankel operators on strongly pseudoconvex domains for symbols that are square integrable only. Recently,Čučković and the second author [ČŞ09, Theorem 3] gave a characterization for compactness of Hankel operators on smooth bounded convex domains in C 2 with symbols smooth up to the boundary. We note that even though they stated their result for smooth domains and smooth symbols on the closure, examination of the proof shows that C 1 -smoothness of the domain and the symbol is sufficient. They proved the following theorem.Theorem (Čučković-Şahutoglu). Let Ω be a C 1 -smooth bounded convex domain in C 2 and φ ∈ C 1 (Ω). Then the Hankel operator H φ is compact on A 2 (Ω) if and only if φ • f is holomorphic for any holomorphic function f : D → bΩ.In this paper we prove a similar result with symbols that are only continuous up to the boundary. The first result in this direction was proven by Le in [Le10]. He showed that for Ω = D n ,
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