A characterization of two-weighted inequalities for maximal, singular operators andtheir commutators in generalized weighted Morrey spaces

In this paper we give a characterization of two-weighted inequalities for maximal, singular operators and their commutators in generalized weighted Morrey spaces $\mathcal{M}^{p,\varphi}_{\omega}(\mathbb{R}^n)$. We prove the boundedness of maximal operator $M$ and maximal commutators $[M,b]$ from the spaces $\mathcal{M}^{p,\varphi_1}_{\omega_1^\delta}(\mathbb{R}^n)$ to the spaces $\mathcal{M}^{p,\varphi_2}_{\omega_2^\delta}(\mathbb{R}^n)$, where $1< p<\infty$, $0<\delta<1$ and $(\omega_1,\omega_2)\in \widetilde{A}_{p}(\mathbb{R}^n)$. We also prove the boundedness of the Calderón--Zygmund singular operators $T$ and their commutators $[b,T]$ from $\mathcal{M}^{p,\varphi_1}_{\omega_1^\delta}(\mathbb{R}^n)$ to $\mathcal{M}^{p,\varphi_2}_{\omega_2^\delta}(\mathbb{R}^n)$. Finally we give generalized weighted Morrey a priori estimates as applications of our results.