Boundedness of vector-valued intrinsic square functions in Morrey type spaces

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogues of intrinsic square functions in the weighted Morrey spaces $L^{p,\kappa}(w)$ when $1\leq p<\infty$, $0<\kappa<1$, and in the generalized Morrey spaces $L^{p,\Phi}$ for $1\le p<\infty$, where $\Phi$ is a growth function on $(0,\infty)$ satisfying the doubling condition.Comment: 14 page

Weighted inequalities for fractional integral operators and linear commutators in the Morrey type spaces

In this paper, we first introduce some new Morrey type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional integral operators $I_\alpha$ in these new Morrey type spaces. Furthermore, the weighted strong type estimate and endpoint estimate of linear commutators $[b,I_{\alpha}]$ formed by $b$ and $I_{\alpha}$ are established. Also we study related problems about two-weight, weak type inequalities for $I_{\alpha}$ and $[b,I_{\alpha}]$ in the Morrey type spaces and give partial results.Comment: 34 pages. arXiv admin note: substantial text overlap with arXiv:1603.0391