Contractivity of Rudin's diagonal restriction operator for $H^p$ of the polydisc

Abstract

We establish that Rudin's diagonal restriction operator is contractive from the Hardy space $H^p$ of the bidisc to the Bergman space $A^p$ of the disc for $2\le p<\infty$, and characterize all cases of equality. We extend the result to polydiscs and obtain applications to polynomial point evaluation among others. The proof relies on an explicit formula, deduced from certain Green's formulas, for the defect between the $p$th powers of the two $L^p$ norms.