Two-weight, weak type norm inequalities for fractional integral operators and commutators on weighted Morrey and amalgam spaces

Abstract

Let $0<\gamma <n$ and $I_{\gamma }$ be the fractional integral operator of order $\gamma$, $I_{\gamma }f(x)={\int }_{{\mathbb{R}}^n}|x-y{|}^{\gamma -n}f(y)dy$, and let $[b,I_{\gamma }]$ be the linear commutator generated by a symbol function $b$ and $I_{\gamma }$, $[b,I_{\gamma }]f(x)=b(x)\cdot I_{\gamma }f(x)-I_{\gamma }(bf)(x)$. This paper is concerned with two-weight, weak type norm estimates for such operators on the weighted Morrey and amalgam spaces. Based on weak-type norm inequalities on weighted Lebesgue spaces and certain $A_p$-type conditions on pairs of weights, we can establish the weak-type norm inequalities for fractional integral operator $I_{\gamma }$ as well as the corresponding commutator in the framework of weighted Morrey and amalgam spaces. Furthermore, some estimates for the extreme case are also obtained on these weighted spaces.

32 pages. arXiv admin note: text overlap with arXiv:1712.01269

References in corpus (2)

• Estimates for fractional integral operators and linear commutators on certain weighted amalgam spaces
• Two-weight, weak type norm inequalities for a class of sublinear operators on weighted Morrey and amalgam spaces