CARLESON MEASURES AND

INTERPOLATING SEQUENCES

1. Introduction

In this paper we consider the analytic Besov spaces Bp (Bn) on the unit ball

B

n

in C n , consisting of those holomorphic functions / on the ball such that

J

\(i-\z\2yf^(Z)\pdxn(z)^,

where m - , d\n (z) = (1 — \z\ J dz is invariant measure on the ball with

dz Lebesgue measure on C

n

, and f^ is the

mth

order complex derivative of

/ . We characterize their Carleson measures (except for p in an exceptional range

2 +

~^~[i00))i

pointwise multipliers and interpolating sequences. We also char-

acterize interpolating sequences for the corresponding pointwise multiplier spaces

MB (Bn) (except for p in the smaller exceptional range 2 -f ^ y , 2 n ). Finally, in

order to obtain the characterization of interpolating sequences for MQ (Bn) in the

difficult range 1 + ^ j p 2, we introduce "holomorphic" Besov spaces HBp (Tn)

on the Bergman trees Tn, and develop the necessary part of the analogous theory

of Carleson measures, pointwise multipliers and interpolating sequences. The main

feature of these holomorphic Besov spaces on Bergman trees is that they provide

a martingale-like analogue of the analytic Besov space on the ball. They also en-

joy properties not found in the actual Besov spaces, such as reproducing kernels

with a positivity property for all p in the range 1 p oc (Lemma 8.11 below).

Various solutions to the problems mentioned here in one dimension can be found

in [Car], [MaSu], [Wu], [Boe] and our earlier paper [ArRoSa]. We remark that

the one-dimensional methods for characterizing multiplier interpolation generalize

for n 1 to prove necessity at most in the ranges 1 p 1 + ~-^ and p 2,

and sufficiency at most in the range p 2n, resulting in a common range of only

p 2n for all n 1.

1.1. History. We begin with an informal discussion of the context in which

our results can be viewed. For more details on this background (at least the part

having to do with Hilbert spaces) we refer to the beautiful recent monograph of K.

Seip [Sei].

The theory of Carleson measures and interpolating sequences has its roots in

Lennart Carleson's 1958 paper [Car], the first of his papers motivated by the corona

problem for the Banach algebra H°° (D) of bounded holomorphic functions in the

l