The Maximal Function of Doubly Commuting Contractions

Abstract

Let D be the open unit disc and T be the unit torus in the complex plane C. Denote by m the normalized Lebesgue measure on T and by m2 =m ⊗ m the normalized Lebesgue measure on the bidimensional torus T2. For a complex separable Hilbert space H, let L2 (T2;H) be the Hilbert space of the measurable functions from T2 to H, with the scalar product $$\left( {{\text{u,v}}} \right)_{{\text{L}}^{\text{2}} \left( {{\text{T}}^{\text{2}} ;{\text{H}}} \right)} = \int\limits_{{\text{T}}^{\text{2}} } {\left( {\text{u}} \right.\left( {\text{w}} \right),{\text{v}}\left( {\text{w}} \right)_{\text{H}} {\text{dm}}_{\text{2}} \left( {\text{w}} \right)} $$ (1.1) .