The Sobolev theorem for anisotropic Riesz-Bessel potentials in Morrey-Bessel spaces
The Morrey-Bessel space \(L^\gamma_{p,\lambda}(\mathbb{R}^n_+)\) is defined as the space of measurable functions \(f\) on \(\mathbb{R}^n_+\) for which the norm \(\|f\|_{L^\gamma_{p,\lambda}}= \sup_{x,t} (t^{-\lambda} \int_{E_t}|T^y f(x)|^p dy)^{1/p}< \infty\), \(E_t\) being some distribution function and \(T^y f(x)\) -- a generalized shift operator. The main result is the inequality \(\|R^\alpha_B f\|_{L^\gamma_{p,\lambda}}\leq C_{p,\lambda}\|f\|_{L^\gamma_{p,\lambda}}\), where \(R_Bf(x)\) is the unisotropic Riesz-Bessel potential of \(f\). No proofs are given.
scientific article; zbMATH DE number 1557748