On the relation of Carleson's embedding and the maximal theorem in the context of Banach space geometry

TL;DR

This paper establishes the equivalence between certain vector-valued Carleson embedding theorems and geometric properties of Banach spaces, linking analytic conditions with Banach space geometry.

Contribution

It proves that the boundedness of a vector-valued maximal operator and the type p property are both necessary and sufficient for the embedding theorems, revealing new equivalences.

Findings

Boundedness of vector-valued maximal operator is necessary and sufficient.

Type p property of Banach space is necessary and sufficient.

New equivalences between analytic and geometric Banach space properties.

Abstract

Hyt\"onen, McIntosh and Portal (J. Funct. Anal., 2008) proved two vector-valued generalizations of the classical Carleson embedding theorem, both of them requiring the boundedness of a new vector-valued maximal operator, and the other one also the type p property of the underlying Banach space as an assumption. We show that these conditions are also necessary for the respective embedding theorems, thereby obtaining new equivalences between analytic and geometric properties of Banach spaces.