# Muckenhoupt weights and maximal function

It is well known that $w \in A_p(\mathbb{R}^n)$ (the Muckenhoupt weight class) if and only $$\int_{\mathbb{R}^n} \mathcal{M} (|f|)^p w(x) \ dx \leq C\int_{\mathbb{R}^n} |f|^p w(x) \ dx.$$

My question is if one takes $M_{<R}$ for some $R>0$, the truncated maximal function defined as $$M_{<R}(|f|)(x) := \sup_{r\leq R} \frac{1}{|B(x,r)|}\int_{B(x,r)}|f|\ dx$$then can improve the weight class? i.e what is the if and only if condition on $u$ such that $$\int_{\mathbb{R}^n} \mathcal{M}_{<R} (|f|)^p u(x) \ dx \leq C\int_{\mathbb{R}^n} |f|^p u(x) \ dx$$holds?

Definitely the above holds if $u \in A_p$. Does it hold for a larger class $\tilde{A}_p \subsetneq A_p$?

• It is known that for any $f \in L^1$, the $M(f)^{\beta}\in A_1$ weight for $\beta \in (0,1)$. The Jones factorization of Muckenhoupt weight says $w \in A_p$ if and only if $w = w_1^{1-p}w_2$ with $w_1,w_2 \in A_1$. In particular, the $L^p$ bounds for the maximal function holds for weights of the form $M(f)^{\beta(1-p)}w_1$ for $\beta \in (0,1)$. By enlarging the weight class (in the context of my main question), can one get maximal bounds for weights of the form $M(f)^{1-p} w_1$? (can I take $\beta =1$?)