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Maximum modulus sets

Thomas Duchamp, Edgar Lee Stout (1981)

Annales de l'institut Fourier

We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain D of dimension N . If Σ b D is a smooth manifold of dimension N and a maximum modulus set, then it admits a unique foliation by compact interpolation manifolds. There is a semiglobal converse in the real analytic case. Two functions in A 2 ( D ) with the same smooth N -dimensional maximum modulus set are analytically related and are polynomially related if a certain homology class in H 1 ( D , R ) vanishes or if D C N is polynomially...

Mixed-norm spaces and interpolation

Joaquín Ortega, Joan Fàbrega (1994)

Studia Mathematica

Let D be a bounded strictly pseudoconvex domain of n with smooth boundary. We consider the weighted mixed-norm spaces A δ , k p , q ( D ) of holomorphic functions with norm f p , q , δ , k = ( | α | k ʃ 0 r 0 ( ʃ D r | D α f | p d σ r ) q / p r δ q / p - 1 d r ) 1 / q . We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces A δ , k p ( D ) and we give results about real and complex interpolation between them. We apply these results to prove that A δ , k p , q ( D ) is the intersection of a Besov space B s p , q ( D ) with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm...

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