We provide geometric conditions on the set of boundary points of infinite type of a smooth bounded pseudoconvex domain in ރ n implying that the ∂-Neumann operator is compact. These conditions are formulated in terms of certain short time flows in suitable complex tangential directions. It is noteworthy that compactness is not established via the known potential theoretic sufficient conditions. Our results generalize to ރ n the ރ 2 results of the second author.
We establish compactness estimates for ∂ M on a compact pseudoconvex CRsubmanifold M of C n of hypersurface type that satisfies the (analogue of the) geometric sufficient conditions for compactness of the ∂-Neumann operator given in [28,20]. These conditions are formulated in terms of certain short time flows in complex tangential directions. * b u, and u (so called 'maximal estimates' hold). * M ) ⊥ ([30], Lemma 4.3). Similarly, (7) says that the canonical solution operator to ∂ * M is compact as an operator from Im(∂ * M ) = ker(∂ M ) ⊥ to ker(∂ * M ) ⊥ = Im(∂ M ). But these two operators are adjoints of each other, so that one is compact if and only if the other is. This completes the proof of Theorem 2.
Abstract. We present a family of weights on the unit disc for which the corresponding weighted Szegö projection operators are irregular on L p spaces. We further investigate the dual spaces of weighted Hardy spaces corresponding to this family.
We construct bounded pseudoconvex domains in C 2 for which the Szegö projection operators are unbounded on L p spaces of the boundary for all p = 2.Mathematics Subject Classification (2010). Primary 32A25. Secondary 32A36, 47B34.
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