BMO spaces on domains of \(R^n\). (English) Zbl 0931.46022
Let \(\Omega\) be an open subset of \(\mathbb{R}^n\) and \(\text{MBO}(\Omega)\) denote the space of functions in \(L^1_{\text{loc}}(\overline\Omega)\) with bounded mean oscillation. The authors’ main result is that if \(\Omega\) is sufficiently regular, then there is a bounded linear extension operator from \(\text{BMO}_1(\Omega)\) into \(\text{BMO}_1(\mathbb{R}^n)\), where \(\text{BMO}_1(\Omega)\) is somewhat smaller than \(\text{BMO}(\Omega)\). An earlier extension theorem was obtained by P. W. Jones [Indiana Univ. Math. J. 29, 41-66 (1980; Zbl 0432.42017)] with a different regularity condition on \(\Omega\) and a different subset of \(\text{BMO}(\Omega)\).
Reviewer: J.V.Whittaker (Vancouver)
MSC:
46E15 | Banach spaces of continuous, differentiable or analytic functions |
47B38 | Linear operators on function spaces (general) |