Interpolating Sequences in the Polydisc

Abstract

ABSTRACT. 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. 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:

Compact composition operators with nonlinear symbols on the H2 space of Dirichlet series

“…Theorem 2 is truly a non-linear extension of the results for linear symbols, however it fails to handle the relatively simple cases (4) ϕ 1 (s) = 9 2 − 2 −s − 3 −s − 2 · 6 −s and ϕ 2 (s) = 13 2 − 4 · 2 −s − 4 · 3 −s + 2 · 6 −s , where "mixed terms" are present. However, the compactness of the associated operators can be decided by our main result.…”

“…At this point we should mention that Theorem 3 does not encompass Theorem 2, and we will return to this point later (see Section 7). However, Theorem 3 allows us to conclude that for the Dirichlet polynomials ϕ given by (4), which have complex dimension and degree equal to 2, the induced composition operators are compact.…”

“…[3, Lin. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] We begin by looking at the coefficients of uv 2 in the real part of Φ(u, v ). Using on the one hand (13) and on the other hand (14) we conclude that γ 0,2 = −6a 3 1 Im(b 2 ) − 6a 3 2 Im(b 1 ) + a 1 a 2 (a 1 + a 2 )Im(c) 8a 1 a 2 .…”

“…, 2(1−|z d (α)|). By Chang's characterization of Carleson measures on the polydisc (see [4] or [6]), it is enough to show that we for all open sets…”

“…Example. Let ϕ 2 (s) = 13/2 − 4 · 2 −s − 4 · 3 −s + 2 · 6 −s as in (4) and let Φ be its minimal Bohr lift. It can be shown that Re Φ(w) = 0 for w ∈ T 2 if and only if w = (1, 1), and…”