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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.

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  • $\begingroup$ Have you checked the book by Hasto? $\endgroup$
    – Piotr Hajlasz
    Commented Apr 10, 2018 at 0:08
  • $\begingroup$ Unfortunately, Hasto's book did not have the right estimates. Eventually I was able to prove the required estimate using their idea's. Please see Theorem 4.8 and Corollary 4.9 in arxiv.org/abs/1707.02535 which was sufficient for my purposes. $\endgroup$
    – Adi
    Commented Apr 11, 2018 at 6:02
  • $\begingroup$ @Adi If you've found an answer to your own question, the best thing to do is to post a separate answer below and click the "checkmark" to the left so that the system marks your question as answered. $\endgroup$
    – j.c.
    Commented Apr 11, 2018 at 8:29

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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 $\int_{B_{2R}} |f(x)|^{p(x)} dx$.

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