Estimates for vector-valued intrinsic square functions and their commutators on certain weighted amalgam spaces

In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we deal with the vector-valued intrinsic square functions, which are given by \begin{equation*} \mathcal S_\gamma(\vec{f})(x) = \Bigg(\sum_{j=1}^\infty \big|\mathcal S_\gamma(f_j)(x) \big|^2\Bigg)^{1/2}, \end{equation*} where $0<\gamma\leq1$ and \begin{equation*} \mathcal S_\gamma (f_j)(x) = \left(\iint_{\Gamma(x)} \Big[\sup_{\varphi\in{\mathcal C}_\gamma} \big|\varphi_t*f_j(y) \big|\Big]^2 \frac{dydt}{t^{n+1}}\right)^{1/2}, \quad j=1,2,\dots. \end{equation*} In his fundamental work, Wilson established strong-type and weak-type estimates for vector-valued intrinsic square functions on weighted Lebesgue spaces. The goal of this paper is to extend his results to these weighted amalgam spaces. Moreover, we define vector-valued analogues of commutators with $BMO(\mathbb R^n)$ functions, and obtain the mapping properties of vector-valued commutators on the weighted amalgam spaces as well. In the endpoint case, we also establish the weighted weak $L\log L$-type estimates for vector-valued commutators in the setting of weighted Lebesgue spaces.