Abstract
LetD be a bounded plane domain (with some smoothness requirements on its boundary). LetBp(D), 1≤p<∞, be the Bergmanp-space ofD. In a previous paper we showed that the “natural projection”P, involving the Bergman kernel forD, is a bounded projection fromLp(D) ontoBp(D), 1<p<∞. With this we have the decompositionLp(D)=Bp(D)⊕Bq⊥(D,p–1+q–=1, 1<p< ∞. Here, we show that the annihilatorBq⊥(D) is the space of allLp-complex derivatives of functions belonging to Sobolev space and which vanish on the boundary ofD. This extends a result of Schiffer for the casep=2. We also study certain operators onLp(D). Especially, we show that Open image in new window , whereI is the identity operator and ℒ is an operator involving the adjoint of the Bergman kernel. Other relationships relevant toBq⊥(D) are studied.
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