Vol. 103, No. 2, 1982

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ISSN: 0030-8730
Carleson measures for functions orthogonal to invariant subspaces

William S. Cohn

Vol. 103 (1982), No. 2, 347–364
Abstract

Let D = {z : |z| < 1} be the unit disk. Suppose φ is an inner function with singular support K and let M = H2 φH2 where H2 is the usual class of functions holomorphic on D. If μ is a positive measure on D, the closed disk, which assigns zero mass to K, then call μ a Carleson measure for M if for a c > 0,

∫
|f|2 dμ ≦ c∥f∥2
2

for all f M. (Here and elsewhere, f2 denotes the H2 norm of an H2 function.) In this paper the Carleson measures for M are characterized for all inner functions φ such that for some 𝜀, 0 < 𝜀 < 1, the set {z : |φ(z)| < 𝜀} is connected.

Mathematical Subject Classification 2000
Primary: 30D55
Secondary: 47B35
Milestones
Received: 9 September 1981
Published: 1 December 1982
Authors
William S. Cohn