# The optimal asymptotic behavior of the coefficient in the Hardy-Littlewood maximal inequality

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 $$n$$?

• Which maximal function? (central or not? using what convex body (cubes? balls?,...), etc.) – fedja Jun 17 at 5:42
• Have you even tried searching (say, in MathSciNet, or even in Google) for, say, "best/optimal constant" and "Hardy–Littlewood"? – Mateusz Kwaśnicki Jun 17 at 12:52

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.