Nehari and Nevanlinna-Pick Problems and Holomorphic Extensions in the Polydisk in Terms of Restricted BMO

A new class BMOr is introduced, motivated by the following extension of the Nehari theorem to d ≥ 1: The norm of a bounded (big) Hankel form B in the d-dimensional torus satisfies ||φ|| ≤ ||B|| ≤ √ d ||φ|| where the norm of φ, the antianalytic symbol of B, is taken in BMOr. This extension in turn provides solutions to Nevanlinna-Pick type interpolation problems in the polydisk, with control of the BMOr norm. Necessary and sufficient conditions for the existence of H2 ∩ BMOr extensions of analytic functions defined on a zero variety V are given, with control of the norm, under general conditions on V.