Approximation and Extension of Functions of Vanishing Mean Oscillation
| Autor: | Almaz Butaev, Galia Dafni |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
010102 general mathematics
Mathematics::Analysis of PDEs A domain Lipschitz continuity 01 natural sciences Omega Combinatorics symbols.namesake Differential geometry Fourier analysis Linear extension Bounded function Norm (mathematics) 0103 physical sciences symbols 010307 mathematical physics Geometry and Topology 0101 mathematics Mathematics |
| Zdroj: | The Journal of Geometric Analysis. 31:6892-6921 |
| ISSN: | 1559-002X 1050-6926 |
| DOI: | 10.1007/s12220-020-00526-8 |
| Popis: | We consider various definitions of functions of vanishing mean oscillation on a domain $$\Omega \subset {{{\mathbb {R}}}^n}$$ . 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 $${{{\mathbb {R}}}^n}$$ , and also extends $$\mathrm{BMO}(\Omega )$$ to $$\mathrm{BMO}({{{\mathbb {R}}}^n})$$ , 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 $$\Omega $$ to functions in $$\mathrm{BMO}({{{\mathbb {R}}}^n})$$ , which is bounded in the $$\mathrm{BMO}$$ 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. |
| Databáze: | OpenAIRE |
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