Characterizations of Arveson's Hardy Space

A Hardy-type space H ² _d in the unit ball B_d of C^d , which was recently introduced by Arveson W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], is appropriate for the operator theory of d-contractions. In this article, it is proved that H ² _d actually coincides with a Hardy-Sobolev space. This yields almost immediately some of the related results obtained in W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], including the facts that H ² _d is not associated with any measure on C ^d ; and that the corresponding algebra of multipliers M ? H ^∞(B_d ) and the inclusion is proper. Finally, a function-theoretic version of von Neumann's inequality for the d-contractions is presented.