Homogeneous fractional integrals on Hardy spaces
Let \(S^{n-1}\) be the unit sphere in the \(n\) dimensional Euclidean space \(\mathbb R^n\). Suppose \(0<\alpha<n\), and let \(\Omega\in L^r(S^{n-1})\) with \(r\geq 1\) be homogeneous of degree zero on \(\mathbb R^n\). The authors treat the following homogeneous fractional integral operator \[ T_{\Omega, \alpha}f(x)=\int_{\mathbb R^n}\frac{\Omega(x-y)}{|x-y|^{n-\alpha}} f(y) dy. \] When \(\Omega\equiv 1\), this is the Riesz potential operator \(I_\alpha\). In the case \(1<p<n/\alpha\), \((L^p, L^q)\) boundedness of \(T_{\Omega, \alpha}\) is already known \((1/q=1/p-\alpha/n)\), as for \(I_\alpha\). As for \(I_\alpha\), \((H^1, L^{n/(n-\alpha)})\), \((H^p,L^q)\) or \((H^p, H^q)\) \((0<p<1)\) boundedness results are known (\(H^p\) is the Hardy space). The authors show, in this paper, that similar results hold for \(T_{\Omega, \alpha}\) under \(L^r\) Dini-type condition on \(\Omega\). The \((H^p,L^q)\) result is as follows: Let \(0<\alpha<1\), \(n/(n+\alpha)\leq p<1\), \(1/q=1/p-\alpha/n\) and \(\Omega\in L^r(S^{n-1})\) with \(r>n/(n-\alpha)\). Suppose \(\int_{0}^{1}\omega_r(\delta)\delta^{-1-\alpha} d \delta<\infty\). Then there exists a \(C>0\) such that \(\|T_{\Omega, \alpha}f\|_{L^q}\leq C\|f\|_{H^p}\). Here, \(\omega_r(\delta)=\sup_{|\rho-I|<\delta}\left(\int_{S^{n-1}} |\Omega(\rho x')-\Omega(x')|^r dx'\right)^{1/r}\), \(\rho\) is a rotation in \(\mathbb R^n\) and \(I\) is the identity operator. Results for \((H^1, L^{n/(n-\alpha)})\) and \((H^p, H^q)\) are also given.
scientific article; zbMATH DE number 1484780