Littlewood-Paley $g^{\ast}_{\lambda,\mu}$-function and Its Commutator on Non-homogeneous Generalized Morrey Spaces

Abstract

Let $\mu$ be a non-negative Radon measure on $\mathbb{R}^{d}$ which may be a non-doubling measure. In this paper, the authors prove that the Littlewood-Paley $g^{\ast}_{\lambda,\mu}$-function is bounded on the generalized Morrey space $\mathcal{L}^{p,\phi}(\mu)$, and also obtain that the commutator $g^{\ast}_{\lambda,\mu,b}$ generated by the Littlewood-Paley function $g^{\ast}_{\lambda,\mu}$ and the regular bounded mean oscillation space $(=$RBMO$)$, which is due to X. Tolsa, is bounded on $\mathcal{L}^{p,\phi}(\mu)$. As a corollary, the authors prove that the commutator $g^{\ast}_{\lambda,\mu,b}$ is bounded on the Morrey space $\mathcal{M}^{p}_{q}(\mu)$ defined by Sawano and Tanaka when we take $\phi(t)=t^{1-\frac{p}{q}}$ with $1<p<q<\infty$.