Area inequality and [formula omitted] norm

Abstract

In this paper, we introduce the norm squares B p for Q p spaces on a hyperbolic Riemann surface R so that B p ( f ) = 1 for 0 ⩽ p ⩽ ∞ if R is the unit disk and f is the identity function, and prove the sharp inequality B p ( f ) ⩽ B q ( f ) for 0 ⩽ q < p ⩽ ∞ . The equality statement is also settled. This is a stronger version of the known nesting property: AD ( R ) = Q 0 ( R ) ⊂ Q q ( R ) ⊂ Q p ( R ) ⊂ Q ∞ ( R ) = CB ( R ) for 0 < q < p < ∞ , where AD ( R ) and CB ( R ) are the Dirichlet space and Bloch-type space, respectively.