I am trying to show an estimate of the following form: Given any $p(x)$ such that $1<p^-\leq p(x) \leq p^+ <\infty$ and $p(\cdot)$ is log-Holder continuous, does there exists an $R\_0$ (depending only on $p(\cdot)$, $n$ and log-Holder continuity of $p(x)$) such that
$$ \int\_{B\_R} M\_{<{2R}} (|f|)^{p(x)}(x) \ dx \leq C \int\_{B\_{2R}} |f(x)|^{p(x)} \ dx + 1$$
holds for every function $f \in L^{p(\cdot)} (B\_{2R})$ and every $R<R\_0$.


Here $M\_{<T}(|f|)(x) := \sup\_{r<T} \frac{1}{|B(x,r)|}\int\_{B(x,r)} |f(y)| \ dy$


EDIT: I have been informed by Peter Hasto that the above estimate is true only under a size restriction on $\int\_{B\_{2R}}|f(x)|^{p(x)} \ dx$.


EDIT: The answer to this question can be found in Theorem 4.8 and Corollary 4.9 of arxiv.org/abs/1707.02535.



Integral form of maximal function estimate on variable exponent spaces
The answer to this question can be found in Theorem 4.8 and Corollary 4.9 of arxiv.org/abs/1707.02535 where a bound of the above form is proved with a size restriction on $\int\_{B\_{2R}} |f(x)|^{p(x)} dx$.


