Estimates for fractional integral operators and linear commutators on certain weighted amalgam spaces (Q2198803)

Summary: 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.