Double Points Local Hardy-Littlewood Maximal Operator

Abstract

A double points local Hardy-Littlewood maximal operator $M_{\left\{a,b\right\},k,\text{loc}}$ is defined and investigated in Euclidean spaces. It is proved that $M_{\left\{a,b\right\},k,\text{loc}}$ is bounded on $L^p\left(w\right)$ when $p>1$ and from $L^1\left(w\right)$ to $L^{1,\infty }\left(w\right)$ with weight function $w\in A_{\left\{a,b\right\},k,\text{loc}}$, the class of double points local $A_p$ weights which is larger than the Muckenhoupt $A_p$ class and the local $A_p$ weights defined by Lin and Stempak.