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It is well-known that for $f \in L^1(\mathbb{R^n})$,$\mu(x \in \mathbb{R^n} | Mf(x) > \lambda) \le \frac{C_n}{\lambda} \int_{\mathbb{R^n}} |f| \mathrm{d\mu}$, where $C_n$ is a constant only depends on $n$.

It is easy to see $C_n \le 2^n$, but how to determine its optimal asymptotic behavior? For example, does $C_n$ bounded in $n$? Is $C_n$ bounded by polynomial in $n$?

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    $\begingroup$ Which maximal function? (central or not? using what convex body (cubes? balls?,...), etc.) $\endgroup$ – fedja Jun 17 at 5:42
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    $\begingroup$ Have you even tried searching (say, in MathSciNet, or even in Google) for, say, "best/optimal constant" and "Hardy–Littlewood"? $\endgroup$ – Mateusz Kwaśnicki Jun 17 at 12:52
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As far as I know the best bound is still $O(n)$ from [1]. For more general maximal functions they get $O(n\log(n))$. See also the historical discussion and results in [2]. Note that in $L^p$ for $p>1$ there is a uniform bound in [1]. Also relevant are [3] and [4].

[1] Stein, Elias M., and Jan-Olov Strömberg. "Behavior of maximal functios in R n for large n." Arkiv för matematik 21, no. 1 (1983): 259-269.

[2] Naor, Assaf, and Terence Tao. "Random martingales and localization of maximal inequalities." Journal of Functional Analysis 259.3 (2010): 731-779.

[3] Bourgain, Jean. "On the Hardy-Littlewood maximal function for the cube." Israel Journal of Mathematics 203, no. 1 (2014): 275-293.

[4] https://msp.org/pjm/2013/263-1/p11.xhtml

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