The authors study the boundedness of the singular integral operator \(T\) and its commutator \([T,\vec b]\) on generalized weighted Morrey spaces \(M_{p,\varphi}(w)\) with the weight function \(w\) belonging to Muckenhoupt’s class \(A_p(\mathbb R^n)\). They find sufficient conditions on the pair \((\varphi_1, \varphi_2)\) with \(\vec{b}\in BMO^m(\mathbb R^n)\) and \(w \in A_p(\mathbb R^n)\) which ensures the boundedness of the operators \(T\) and \([T,\vec b]\) from \(M_{p,\varphi_1}(w)\) to \(M_{p,\varphi_2}(w)\) for \(1< p<\infty\).
Let \(\omega\) be a non-negative and non-decreasing function on \((0,\infty)\). We say that \(\omega\) satisfies the Dini condition, if \(\int_0^1\omega(t)dt/t<\infty\). A measurable function \(K(\cdot,\cdot)\) on \(\mathbb R^n\times \mathbb R^n\) is said to be an \(\omega\)-type Calderón-Zygmund kernel if it satisfies \(|K(x, y)|\le C |x-y|^{-n}\) and for all distinct \(x, y \in \mathbb R^n\), and all \(z\) with \(2|x-z| < |x-y|\), there exists positive constants \(C\) such that \(|K(x, y)-K(z, y)| + |K(y, x)-K(y, z)|\le C\omega(|x-z|/|x-y|)|x-y|^{-n}\).
One of the main theorems is the following. Let \(1\le p<\infty\), \(w\in A_p(\mathbb R^n)\), \(T\) be a singular integral operator with \(\omega\)-type Calderón-Zygmund kernel function \(K\), and \((\varphi_1, \varphi_2)\) satisfy the condition \(\int^\infty_r \frac{ \operatorname{ess\,\,inf}_{t<s<\infty} \varphi_1(x, s)w(B(x, s))^{1/p}}{w(B(x, t))^{1/p}}\le C\varphi_2(x, r)\), where \(C\) does not depend on \(x\) and \(r\). Then the operator \(T\) is bounded from \(M_{p,\varphi_1}(w)\) to \(M_{p,\varphi_2}(w)\) for \(p > 1\) and from \(M_{p,\varphi_1}(w)\) to weak \(WM_{p,\varphi_2}(w)\) for \(p = 1\).
They also state several related results.