Parameter θ -Type Marcinkiewicz Integral on Nonhomogeneous Weighted Generalized Morrey Spaces

Abstract

Let $\left(\mathcal{X},d,\mu \right)$ be a nonhomogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. In this setting, the author proves that parameter $\theta$ -type Marcinkiewicz integral ${\mathcal{M}}_{\theta }^{\rho }$ is bounded on the weighted generalized Morrey space $L^{p,\varphi ,\tau }\left(\omega \right)$ for $p\in \left(1,\infty \right)$ . Furthermore, the boudedness of ${\mathcal{M}}_{\theta }^{\rho }$ on weak weighted generalized Morrey space $W{L}^{p,\varphi ,\tau }\left(\omega \right)$ is also obtained.