Vol. 99, No. 1, 1982

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The construction of certain BMO functions and the corona problem

Akihito Uchiyama

Vol. 99 (1982), No. 1, 183–204
Abstract

In Euclidean space Rd, let I denote any cube with sides parallel to the axes and write |I| for the measure of I. A real valued locally integrable function f(x) on Rd has bounded mean oscillation, f BMO, if

      ∫
supinf  |f(x)− c|dx∕|I| = ∥f∥   < ∞.
I c∈R I                   BMO

Our result is the following.

Theorem 1. Let λ > 1. Let E1,,EN Rd be measurable sets such that

 min |I ∩ Ej|∕|I| < 2−2dλ                (1.1)
1≤j≤N

for any I. Then, there exist functions {fj(x)}j=1N such that

 N
∑  fj(x) ≡ 1,                      (1.2)
j=1

0 ≦ fj(x) ≦ 1, 1 ≦ j ≦ N,                 (1.3)

fj(x) = 0 a.e. on Ej, 1 ≦ j ≦ N,             (1.4)

∥fj|BMO ≦ c1(d,N )∕λ,  1 ≦ j ≦ N.             (1.5)

Converely, if there exist {fj(x)}j=1N that satisfy (1.2)–(1.4) and

∥fj|BMO ≦ c2(d,N )∕λ,  1 ≦ j ≦ N,             (1.6)

then (1.1) holds.

Mathematical Subject Classification 2000
Primary: 42B10
Secondary: 46E99, 46J15
Milestones
Received: 20 April 1980
Published: 1 March 1982
Authors
Akihito Uchiyama