Estimates for fractional integral operators and linear commutators on certain weighted amalgam spaces

In this paper, we first introduce some new classes of weighted amalgam spaces. Then we give the weighted strong-type and weak-type estimates for fractional integral operators $I_\gamma$ on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators $[b,I_{\gamma}]$ generated by $b$ and $I_{\gamma}$ are established as well. In addition, we are going to study related problems about two-weight, weak type inequalities for $I_{\gamma}$ and $[b,I_{\gamma}]$ on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove norm inequalities involving fractional maximal operator $M_{\gamma}$ and generalized fractional integrals $\mathcal L^{-\gamma/2}$ in the context of weighted amalgam spaces, where $0<\gamma<n$ and $\mathcal L$ is the infinitesimal generator of an analytic semigroup on $L^2(\mathbb R^n)$ with Gaussian kernel bounds.