One-sided Muckenhoupt weights and one-sided weakly porous sets in R

In this work, we introduce the geometric concept of one-sided weakly porous sets in the real line and show that a set $E\subset R$ satisfies $d{(\cdot ,E)}^{-\alpha }\in A_1^{+}\left(R\right)\cap L_{\text{loc}}^1\left(R\right)$ for some $\alpha >0$ if and only if E is right-sided weakly porous. Furthermore, we find that the property of being both left-sided and right-sided weakly porous is equivalent to the recent weakly porous condition discussed in the bibliography, which, in turn, was previously found to be intimately related to the usual class of Muckenhoupt weights $A_1$.