Geodesic
BMO-scale of distribution on $\mathbb {R}^n$
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 505-516 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 32A37, 46E30, 46F05
Keywords: $\mathop {\rm BMO}$; $\mathop {\rm VMO}$; John and Niereberg; Bessel potential 
@article{CMJ_2008_58_2_a15,
     author = {Castillo, Ren\'e Erl{\'\i}n and Fern\'andez, Julio C. Ramos},
     title = {BMO-scale of distribution on $\mathbb {R}^n$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {505--516},
     year = {2008},
     volume = {58},
     number = {2},
     mrnumber = {2411106},
     zbl = {1171.46310},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a15/}
}
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Castillo, René Erlín; Fernández, Julio C. Ramos. BMO-scale of distribution on $\mathbb {R}^n$. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 505-516. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a15/