Approximation and Extension of Functions of Vanishing Mean Oscillation
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Approximation and Extension of Functions of Vanishing Mean Oscillation Almaz Butaev1 · Galia Dafni2 Received: 4 December 2019 / Accepted: 21 September 2020 © Mathematica Josephina, Inc. 2020
Abstract We consider various definitions of functions of vanishing mean oscillation on a domain ⊂ Rn . If the domain is uniform, we show that there is a single extension operator which extends functions in these spaces to functions in the corresponding spaces on Rn , and also extends BMO() to BMO(Rn ), generalizing the result of Jones. Moreover, this extension maps Lipschitz functions to Lipschitz functions. Conversely, if there is a linear extension map taking Lipschitz functions with compact support in to functions in BMO(Rn ), which is bounded in the BMO norm, then the domain must be uniform. In connection with these results we investigate the approximation of functions of vanishing mean oscillation by Lipschitz functions on unbounded domains. Keywords Bounded mean oscillation · Vanishing mean oscillation · Continuous mean oscillation · Extension theorems · Uniform domains Mathematics Subject Classification 42B35 · 46E30
Dedicated to the memory of E. M. Stein. Almaz Butaev was partially supported by a PIMS postdoctoral Fellowship at the University of Calgary and the Natural Sciences and Engineering Research Council (NSERC) of Canada. Galia Dafni was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Centre de recherches mathématiques (CRM) and the Fonds de recherche du Québec-Nature et technologies (FRQNT).
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Galia Dafni [email protected] Almaz Butaev [email protected]
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Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada
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Department of Mathematics and Statistics, Concordia University, Montreal, QC H3G 1M8, Canada
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A. Butaev, G. Dafni
1 Introduction Let f be a real-valued function defined on some subset ⊂ Rn . It is a natural question to ask how to extend f to a function on Rn while preserving some of its properties. Those properties can be described by requiring that f belong to some function space. Most trivially, bounded functions on any set can be immediately extended to bounded functions on Rn . Less trivially, Lipschitz continuous functions on any set can be extended to Lipschitz functions on Rn : the McShane–Whitney theorem states that an L-Lipschitz function on a nonempty set can be extended to an L-Lipschitz function F on Rn (see [17]) or more generally on a metric measure space (see [18]), with F = f on , for example by setting F(x) = inf{ f (y) + L|x − y| : y ∈ }. Looking at smoother functions, the problem of extending C m functions from closed sets to Rn crucially depends on how differentiable functions are defined on a closed set. Considering them as jets, Whitney proved the extension theorem introducing his famous decomposition (see e.g. Chapter VI in [29] or Chapter II in [5]). However, defining C m functions on a closed set as traces of C m (Rn ) functions to that set makes th
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