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The (uncentered) Hardy-Littlewood maximal function $M(f)$ of (a locally integrable) function $f$ on $\mathbb{R}^{n}$ is defined by the rule $M(f)(x)=\sup_{\delta>0,\left|y-x\right|<\delta} \text{Avg}_{B(y,\delta)} \left|f\right|$, where $\text{Avg}_{B(y,\delta)} \left|f\right| = \int_{\left|z\right|<\delta} f(y-z) dz$.

The following results regarding the (uncentered) Hardy-Littlewood maximal function are well-known and can be found in many basic texts on analysis (e.g. Loukas Grafakos' "Classical Fourier Analysis", Chapter 2, pages 78-81):

  • The Hardy-Littlewood maximal function is a bounded operator from $L^1(\mathbb{R}^n)$ to $L^{1,\infty}(\mathbb{R}^n)$ (i.e., weak $L^1$) of norm at most $3^n$ ($n$ is the dimension of the Euclidean space).

  • Since the Hardy-Littlewood maximal function is also a bounded operator from $L^{\infty}(\mathbb{R}^n)$ to itself with norm at most $1$ (this is clear), we can apply the Marcinkiewicz interpolation theorem to conclude that for all $1 < p < \infty$, the operator norm of the Hardy-Littlewood maximal function is at most $2\left(\frac{p}{p-1}\right)^{\frac{1}{p}}3^{\frac{n}{p}}$. In fact, there is a slightly better bound: $\frac{p}{p-1}3^{\frac{n}{p}}$.

  • The bound given above grows exponentially with the dimension $n$ (if $p$ is fixed). It is a fact that it cannot be improved to a bound that does not grow exponentially with the dimension $n$.

My questions:

Is an exact value for the norm of the (uncentered) Hardy-Littlewood maximal function, viewed as a bounded operator from $L^p$ to itself ($1<p<\infty$), known? If so, what is it?

Also, what is the norm of the Hardy-Littlewood maximal function when it is viewed as an operator from $L^1$ to weak $L^1$ (if it is known)?

Are the answers to the analogous questions regarding the centered Hardy-Littlewood maximal function known?

I apologize if this question is too basic. It seems like a fairly simple question but it is not clear (at least to me) how to solve it.

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2 Answers

I just searched in google, "best constants for uncentered maximal functions", by Grafakos and Smith. "the best constants for the centered H-L maximal inequality", by AD Melas.(1-D, weak type (1,1)). maybe you can find more results following this two.

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A few comments (a) the second author of the first paper you refer to is Montgomery-Smith, not just Smith. (b) The sharp constants computed by Grafakos and Montgomer-Smith for the uncentered operator, and the sharp constant by Melas, are both results in 1 dimension. Less is known for higher dimensions; though there are known improvements over the exponential bounds for the centered maxmimal functions. The classic paper in that direction is the 1983 paper of Eli Stein and J. Stromberg, though I'm sure there must be other references since then. –  Willie Wong Jul 6 '11 at 13:50
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Uncenteed maximal operator is proven to be unbounded as $L^p \to L^p$ for general norm (see L. Grafakos and J. Kinnunen, "Sharp inequalities for maximal functions associated with general measures") Actually the same result is for $L^1 \to L^{1,\infty}$ (ADAM OSEKOWSKI "BEST CONSTANTS IN THE WEAK-TYPE ESTIMATES FOR UNCENTERED MAXIMAL OPERATORS"). In those papers one can find some results regarding strong maximal function.

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