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Keywords:
$\mathop {\rm BMO}$; $\mathop {\rm VMO}$; John and Niereberg; Bessel potential
$\mathop {\rm BMO}$; $\mathop {\rm VMO}$; John and Niereberg; Bessel potential
Summary:
Let $S^{\prime }$ be the class of tempered distributions. For $f\in S^{\prime }$ we denote by $J^{-\alpha }f$ the Bessel potential of $f$ of order $\alpha $. We prove that if $J^{-\alpha }f\in \mathop {\mathrm BMO}$, then for any $\lambda \in (0,1)$, $J^{-\alpha }(f)_\lambda \in \mathop {\mathrm BMO}$, where $(f)_\lambda =\lambda ^{-n}f(\phi (\lambda ^{-1}\cdot ))$, $\phi \in S$. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order $\alpha >0$ belongs to the $\mathop {\mathrm VMO}$ space.
References:
[1] F. John and L. Nirenberg: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961), 415–426. DOI 10.1002/cpa.3160140317 | MR 0131498
[2] D. Sarason: Functions of bounded mean oscillation. Trans. Amer. Math. Soc. 201 (1975), 391–405. MR 0377518
[3] E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princenton University Press, Princenton, NJ, 1970. MR 0290095 | Zbl 0207.13501
[4] W. R. Wade: An introduction to Analysis, 2nd ed. Prentice Hall, NJ, 2000. Zbl 0951.26001
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