Boundedness of homogeneous fractional integral operator on Morrey space

Abstract

Abstract For 0 < α < n $0<\alpha<n$ , the homogeneous fractional integral operator T Ω , α $T_{\Omega,\alpha}$ is defined by T Ω , α f ( x ) = ∫ R n Ω ( x − y ) | x − y | n − α f ( y ) d y . $$T_{\Omega,\alpha}f(x)= \int_{{\Bbb {R}}^{n}}\frac{\Omega (x-y)}{\vert x-y\vert ^{n-\alpha}}f(y)\,dy. $$ In this paper we prove that if Ω satisfies some smoothness conditions on S n − 1 $S^{n-1}$ , then T Ω , α $T_{\Omega,\alpha}$ is bounded from L λ α , λ ( R n ) $L^{\frac{\lambda}{\alpha },\lambda}({\Bbb {R}}^{n})$ to BMO ( R n ) $\operatorname {BMO}({\Bbb {R}}^{n})$ , and from L p , λ ( R n ) $L^{p,\lambda}({\Bbb {R}}^{n})$ ( λ α < p < ∞ $\frac{\lambda}{\alpha}< p<\infty$ ) to a class of the Campanato spaces L l , λ ( R n ) $\mathcal{L}_{l,\lambda }({\Bbb {R}}^{n})$ , respectively.

Keywords

• Morrey space
• Campanato space
• BMO space
• homogeneous fractional integral operator