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The Hardy-Littlewood-Bessel maximal functions (B-maximal functions), Morrey-Bessel and BMO-Bessel spaces were introduced and studied in [6]. In the present paper, we study the anisotropic Riesz-Bessel potential (B-potential) in the Morrey-Bessel and BMO-Bessel spaces. We obtain a theorem analogous to the Sobolev theorem, for the anisotropic Riesz-Bessel potential in Morrey-Bessel spaces. We introduce a metric characteristic Op,? in the space of locally integrable functions and establish estimates connecting the characteristics of the image and preimage of the corresponding integral transform. These estimates are of independent interest. Moreover, they are used for the investigation of integral operators in different scales of Banach function spaces, in particular, in weighted L p ?-spaces. The results seem to be new even in the isotropic case.

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In this paper, in order to consider the problems of relative width on ℝd, we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d ))$$ \end{document}
hold, where U(W 2 α) is the Riesz potential or Bessel potential of the unit ball in L 2(ℝk) and the notations
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\overline K _\sigma$$ \end{document}
(·, ·,L 2(ℝd)) and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\overline d _\sigma$$ \end{document}
(·, L 2(ℝd)) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )),$$ \end{document}
where 0 × β × α.

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