A two-weight estimate for a class of fractional integral operators with rough kernel (Q2575984)

The author studies \((L^{p}, L^{q})\) mapping properties with some weight for the rough fractional integral operator. More precisely, let \(\Omega \in L^{s}(\mathbb{S}^{n-1})(s >1)\) be homogeneous of degree zero in \(\mathbb{R}^{n}\) with zero mean value on \(\mathbb{S}^{n-1}\). The operator \(T_{\Omega, \alpha}^{A}\) is defined by \[ T_{\Omega, \alpha}^{A}f(x) = \int_{\mathbb{R}^{n}} \frac{R_{m}(A,x,y)}{| x-y| ^{n-\alpha+m-1}} \Omega(x-y) \, f(y) \, dy, \] and the related maximal operator \(M_{\Omega, \alpha}^{A}\) by \[ M_{\Omega, \alpha}^{A}f(x) = \sup_{r>0} \frac{1}{r^{n-\alpha+m-1}} \int_{| x-y| < r} | R_{m}(A,x,y)| | \Omega(x-y) f(y)| \, dy, \] where \(0 < \alpha < n\) and \[ R_{m}(A,x,y)= A(x)- \sum_{| \gamma| <m} \frac{1}{\gamma !} D^{\gamma}A(y)(x-y)^{\gamma}, \] where \(D^{\gamma}A \in L^{r}(\mathbb{R}^{n})\) \((1 < r \leq \infty )\) or \(D^{\gamma}A \in \mathrm{BMO}(\mathbb{R}^{n})\) for \(| \gamma| = m-1 (m \geq 1)\). Here a pair \((u(x), v(x))\) of nonnegative locally integrable functions is said to belong to \(A^{*}(p,q)\) \((1 < p,q < \infty)\) if there is a constant \(C > 0\) such that \[ \sup_{Q} \big( \frac{1}{| Q| }\int_{Q} u(x)^{q} \, dx \big)^{1/q} \, \big( \frac{1}{| Q| }\int_{Q} v(x)^{-p'} \, dx \big)^{1/p'} \leq C < \infty. \] He shows that the rough fractional integral operator \(T_{\Omega, \alpha}^{A}\) and the related maximal operators \(M_{\Omega, \alpha}^{A}\) are bounded from \(L^{p}(v^{p})\) to \(L^{q}(u^{p})\) with the weight pair \((u,v)\) for somw \(s, r, p, q\) and \(u, v\) in \(A^{*}(p,q)\).