Two-weighted inequalities for maximal, singular integral operators and their commutators in \(G\mathcal{M}_{p, \theta, \omega, \varphi}(\mathbb{R}^n)\) spaces (Q6606332)

Given \(0<p\le\infty\), \(0<\theta\le\infty\), and non-negative measurable functions \(\omega\) on \((0,\infty)\) and \(\varphi\) on \(\mathbb R^n\), respectively, the weighted global Morrey-type space \(G\mathcal M_{p,\theta,\omega,\varphi}(\mathbb R^n)\) is defined. Then, two-weight inequalities are proved for the centered Hardy-Littlewood maximal operator \(M\) on \(\mathbb R^n\), a singular integral operator \(T\) on \(\mathbb R^n\), and the commutators \([M,b]\) and \([T,b]\), where \(b\in BMO(\mathbb R^n)\). These inequalities establish boundedness of \(M\), \(T\), \([M,b]\) and \([T,b]\), between the spaces \(G\mathcal M_{p,\theta_1,\omega_1,\varphi_1}(\mathbb R^n)\) and \(G\mathcal M_{p,\theta_2,\omega_2,\varphi_2}(\mathbb R^n)\), \(1<p<\infty\), under suitable necessary conditions imposed on \(\theta_1\), \(\theta_2\), \(\omega_1\), \(\omega_2\), and \(\varphi_1\), \(\varphi_2\). As applications, weighted global Morrey-type a priori estimates and a priori estimates for non-divergent elliptic equations in \(G\mathcal M_{p,\theta,\omega,\varphi}(\mathbb R^n)\) spaces are given.