Boundedness of intrinsic square functions and commutators on generalized central Morrey spaces

In this paper, the authors establish the boundedness for a large class of intrinsic square functions $\mathcal{G}_{\alpha}$, $g_{\alpha}$, $g^{\ast}_{\tilde{\lambda},\alpha}$ and their commutators $[b,\mathcal{G}_{\alpha}]$, $[b,g_{\alpha}]$ and $[b,g^{\ast}_{\tilde{\lambda},\alpha}]$ generated with $\lambda$-central $BMO$ functions $b\in CBMO^{p,\lambda}(\mathbb{R}^{n})$ on generalized central Morrey spaces $\mathcal{B}^{q,\varphi}(\mathbb{R}^{n})$ for $1<q<\infty,0<\alpha\leq1$, respectively. All of the results are new even on the central Morrey spaces $\mathcal{B}^{q,\lambda}(\mathbb{R}^{n})$.