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 ${\mathcal{G}}_{\alpha }$, $g_{\alpha }$, $g_{\tilde{\lambda },\alpha }^{\ast }$ and their commutators $[b,{\mathcal{G}}_{\alpha }]$, $[b,g_{\alpha }]$ and $[b,g_{\tilde{\lambda },\alpha }^{\ast }]$ generated with $\lambda$-central $BMO$ functions $b\in CBM{O}^{p,\lambda }({\mathbb{R}}^n)$ on generalized central Morrey spaces ${\mathcal{B}}^{q,\phi }({\mathbb{R}}^n)$ for $1<q<\mathrm{\infty },0<\alpha \le 1$, respectively. All of the results are new even on the central Morrey spaces ${\mathcal{B}}^{q,\lambda }({\mathbb{R}}^n)$.