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Approximation in the Zygmund and Hölder classes on $\mathbb {R}^n$

Published online by Cambridge University Press:  13 September 2021

Eero Saksman
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, Helsinki FIN-00014, Finland e-mail: Eero.Saksman@helsinki.fi
Odí Soler i Gibert*
Affiliation:
Institut für Mathematik, Universität Würzburg, Würzburg 97074, Germany

Abstract

We determine the distance (up to a multiplicative constant) in the Zygmund class $\Lambda _{\ast }(\mathbb {R}^n)$ to the subspace $\mathrm {J}_{}(\mathbf {bmo})(\mathbb {R}^n).$ The latter space is the image under the Bessel potential $J := (1-\Delta )^{{-1}/2}$ of the space $\mathbf {bmo}(\mathbb {R}^n)$ , which is a nonhomogeneous version of the classical $\mathrm {BMO}$ . Locally, $\mathrm {J}_{}(\mathbf {bmo})(\mathbb {R}^n)$ consists of functions that together with their first derivatives are in $\mathbf {bmo}(\mathbb {R}^n)$ . More generally, we consider the same question when the Zygmund class is replaced by the Hölder space $\Lambda _{s}(\mathbb {R}^n),$ with $0 < s \leq 1$ , and the corresponding subspace is $\mathrm {J}_{s}(\mathbf {bmo})(\mathbb {R}^n)$ , the image under $(1-\Delta )^{{-s}/2}$ of $\mathbf {bmo}(\mathbb {R}^n).$ One should note here that $\Lambda _{1}(\mathbb {R}^n) = \Lambda _{\ast }(\mathbb {R}^n).$ Such results were known earlier only for $n = s = 1$ with a proof that does not extend to the general case.

Our results are expressed in terms of second differences. As a by-product of our wavelet-based proof, we also obtain the distance from $f \in \Lambda _{s}(\mathbb {R}^n)$ to $\mathrm {J}_{s}(\mathbf {bmo})(\mathbb {R}^n)$ in terms of the wavelet coefficients of $f.$ We additionally establish a third way to express this distance in terms of the size of the hyperbolic gradient of the harmonic extension of f on the upper half-space $\mathbb {R}^{n +1}_+$ .

Type
Article
Copyright
© Canadian Mathematical Society 2021

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Footnotes

First author is supported by the Finnish Academy grant 1309940. Second author is supported by the Generalitat de Catalunya grant 2017 SGR 395, the Spanish Ministerio de Ciencia e Innovación projects MTM2014-51824-P and MTM2017-85666-P, and the European Research Council project CHRiSHarMa no. DLV-682402.

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