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 $\tilde{A}_p \subsetneq A_p$?
yes, it holds for a larger class. Instead of requiring the reverse H"older inequality to hold on every balls, you just have to check the reverse inequality on balls with radius