Interpolating Sequences in the Polydisc

Let H°°(Dn) denote the set of all bounded analytic functions defined on the polydisc D" of Cn. In this note, we give a sufficient condition for sequences of points in £>" to be interpolating sequences for H°°(Dn). We also discuss some conditions for interpolation of general domains. Let G(X) be the set of all bounded continuous functions on a compact set X and let A C C(X) be a uniform algebra equipped with the sup norm. We say that a sequence {a0} of points in X is an interpolating sequence for A if A \ {aj} = l°°. In other words, {aj} is an interpolating sequence for A if whenever given a bounded sequence {aj}, there is a function / G A such that f(aj) = aj. In the case when X = D = the unit disc in C and A — H°° (D) = all the bounded analytic functions on D, Carleson proved the following famous theorem (see [3]). THEOREM (CARLESON). The following are equivalent: (a) There is a constant 6 > 0 such that TT Qfc ~ aj >6>0 for all j. (b) {aj} is an interpolating sequence for H°°(D). (c) For k ^ j, \(ak — aj)/(l — djak)\ > a > 0 for some a, and the measure » = '52(l-\aj\)6ai i is a Carleson measure on D, when Saj is the Dirac measure at aj. If to each point aj we associate an arc Ij in dD centered at a3/\aj\ with length 2(1 — \aj\), then the second part of condition (c) indicates that there is a constant G such that for all arcs I in 3D. (Here \E\ denotes the Lebesgue measure of the set E.) The purpose of this note is to study the analogue of these conditions on the polydisc Dn of Cn. On the same topic, the readers are referred to earlier works of Kronstadt [7] and E. Amar [1]. Our approach to the problem is quite different from theirs. First we recall some definitions: Received by the editors January 21, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 42B30; Secondary 41A05.