Boundedness of sublinear operators with rough kernels on weighted Morrey spaces (Q1951086)

Let \(\Omega\) be a function over the unit sphere \(S^{n-1}\) of \(\mathbb{R}^n (n\geq 2)\), let \(D_k=\{x\in \mathbb{R}^n:|x|\leq 2^k \}\) and \(A_k=D_k\diagdown D_{k-1}\). Suppose \(\mathcal{F}_{\Omega}\) is a sublinear operator satisfying \[ |\mathcal{F}_{\Omega}f(x)|\leq C |x|^{-n}\int_{\mathbb{R}^n}|\Omega(x-y)f(y)|dy, \] when \( \mathrm{supp } f\subseteq A_k\) and \(|x|\geq 2^{k+1}\) with \(k\in \mathbb{Z}\) and \[ |\mathcal{F}_{\Omega}f(x)|\leq C 2^{-kn}\int_{\mathbb{R}^n}|\Omega(x-y)f(y)|dy, \] when \( \mathrm{supp } f\subseteq A_k\) and \(|x|\leq 2^{k-1}\) with \(k\in \mathbb{Z}\), respectively. The authors obtain the boundedness of the operator \(\mathcal{F}_{\Omega}\) with rough kernels on weighted Morrey spaces under generic size conditions. Applications to the corresponding commutators formed by the operator \(\mathcal{F}_{\Omega}\) and BMO functions are also obtained.