A continuity theorem for generalized Riesz potentials
Let \(0< \alpha< n\), \(1/r< 1-{\alpha \over n}\) and let \(k\in L^r (S_{n-1})\), where \(S_{n-1}\) is the unit sphere in \(\mathbb{R}^n\). The generalized Riesz kernel \(K_\alpha\) is given by \(K_\alpha (x)= k ({x\over {|x|}} )|x|^{\alpha -n}\), \(x\in \mathbb{R}^n- \{0\}\). The author proves that: (i) If \(1<p< n/\alpha\) and \(1/q= 1/p- \alpha/n\), then the map \(f\to K_\alpha *f\) is continuous from \(L^p\) into \(L^q\). (ii) The map \(f\to K_\alpha *f\) is of weak type \((1,n/ (n-\alpha))\).
scientific article; zbMATH DE number 720417