The Sobolev theorem for the Riesz \(B\)-potentials
The author announces the following theorems (without proof). Theorem: Assume that \(0<\alpha <n+|\gamma|\), \(1<p< q<\infty\), \(0\leq \lambda\leq n+|\gamma|\). Then there exists a positive constant \(C_p\) such that \[ \|R^\alpha_B f\|_{L^\gamma_{p,\lambda}} \leq C_{p,\lambda} \|f\|_{L^\gamma_{p,\lambda}} \quad\text{if} \quad {1\over p}-{1\over q}= {\alpha\over n+ |\gamma|} \] and \[ \|R^\alpha_B f\|_{*,B}\leq C_p \|f\|_{p, \gamma}\quad \text{if} \quad\alpha= {n+|\gamma|\over p}. \] Theorem: Assume that \(0<\alpha <n+|\gamma|\). Then \(\widetilde R^\alpha_B\) is a bounded operator from \(L^\gamma_{l,n+ |\gamma |-\alpha}\) to \(\text{BMO}_B (\mathbb{R}^n_+)\). Proofs are given in Anal. Math. 26, 99-118 (2000; Zbl 0973.47048).
scientific article; zbMATH DE number 1557251