ABSTRACT
Our goal in this chapter is to continue the study begun in Section 3.5 of boundedness and compactness of Cφ on function spaces in BN when N > 1. The Carleson measure criterion (Theorem 3.35) continues to underlie the results in this chapter, but our focus will be to obtain deeper consequences of this characterization than those in Section 3.5. We begin with a compactness question which is motivated by one of the earliest results on compactness in one variable: if φ(D) is contained in a nontangential approach region in D, then Cφ is compact on HP(D) for all p < oo (Proposition 3.25). We will consider Cφ when φ(BN) is contained in a Koranyi approach region https://www.w3.org/1998/Math/MathML"> Γ ( ζ , α ) = { z ∈ B N : | 1 − 〈 z , ζ 〉 | < α 2 ( 1 − | z | 2 ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315139920/f18e986f-6fad-4b68-b3d6-a7462b0271e1/content/eq1690.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
