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If ${f: {\bf R}^d \rightarrow {\bf C}}$ is a locally integrable function, we define the Hardy-Littlewood maximal function ${Mf: {\bf R}^d \rightarrow {\bf C}}$ by the formula

$\displaystyle Mf(x) := \sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)|\ dy,$

where ${B(x,r)}$ is the ball of radius ${r}$ centred at ${x}$, and ${|E|}$ denotes the measure of a set ${E}$. The Hardy-Littlewood maximal inequality asserts that

$\displaystyle |\{ x \in {\bf R}^d: Mf(x) > \lambda \}| \leq \frac{C_d}{\lambda} \|f\|_{L^1({\bf R}^d)} \ \ \ \ \ (1)$

for all ${f\in L^1({\bf R}^d)}$, all ${\lambda > 0}$, and some constant ${C_d > 0}$ depending only on ${d}$. By a standard density argument, this implies in particular that we have the Lebesgue differentiation theorem

$\displaystyle \lim_{r \rightarrow 0} \frac{1}{|B(x,r)|} \int_{B(x,r)} f(y)\ dy = f(x)$

for all ${f \in L^1({\bf R}^d)}$ and almost every ${x \in {\bf R}^d}$. See for instance my lecture notes on this topic.

By combining the Hardy-Littlewood maximal inequality with the Marcinkiewicz interpolation theorem (and the trivial inequality ${\|Mf\|_{L^\infty({\bf R}^d)} \leq \|f\|_{L^\infty({\bf R}^d)}}$) we see that

$\displaystyle \|Mf\|_{L^p({\bf R}^d)} \leq C_{d,p} \|f\|_{L^p({\bf R}^d)} \ \ \ \ \ (2)$

for all ${p > 1}$ and ${f \in L^p({\bf R}^d)}$, and some constant ${C_{d,p}}$ depending on ${d}$ and ${p}$.

The exact dependence of ${C_{d,p}}$ on ${d}$ and ${p}$ is still not completely understood. The standard Vitali-type covering argument used to establish (1) has an exponential dependence on dimension, giving a constant of the form ${C_d = C^d}$ for some absolute constant ${C>1}$. Inserting this into the Marcinkiewicz theorem, one obtains a constant ${C_{d,p}}$ of the form ${C_{d,p} = \frac{C^d}{p-1}}$ for some ${C>1}$ (and taking ${p}$ bounded away from infinity, for simplicity). The dependence on ${p}$ is about right, but the dependence on ${d}$ should not be exponential.

In 1982, Stein gave an elegant argument (with full details appearing in a subsequent paper of Stein and Strömberg), based on the Calderón-Zygmund method of rotations, to eliminate the dependence of ${d}$:

Theorem 1 One can take ${C_{d,p} = C_p}$ for each ${p>1}$, where ${C_p}$ depends only on ${p}$.

The argument is based on an earlier bound of Stein from 1976 on the spherical maximal function

$\displaystyle M_S f(x) := \sup_{r>0} A_r |f|(x)$

where ${A_r}$ are the spherical averaging operators

$\displaystyle A_r f(x) := \int_{S^{d-1}} f(x+r\omega) d\sigma^{d-1}(\omega)$

and ${d\sigma^{d-1}}$ is normalised surface measure on the sphere ${S^{d-1}}$. Because this is an uncountable supremum, and the averaging operators ${A_r}$ do not have good continuity properties in ${r}$, it is not a priori obvious that ${M_S f}$ is even a measurable function for, say, locally integrable ${f}$; but we can avoid this technical issue, at least initially, by restricting attention to continuous functions ${f}$. The Stein maximal theorem for the spherical maximal function then asserts that if ${d \geq 3}$ and ${p > \frac{d}{d-1}}$, then we have

$\displaystyle \| M_S f \|_{L^p({\bf R}^d)} \leq C_{d,p} \|f\|_{L^p({\bf R}^d)} \ \ \ \ \ (3)$

for all (continuous) ${f \in L^p({\bf R}^d)}$. We will sketch a proof of this theorem below the fold. (Among other things, one can use this bound to show the pointwise convergence ${\lim_{r \rightarrow 0} A_r f(x) = f(x)}$ of the spherical averages for any ${f \in L^p({\bf R}^d)}$ when ${d \geq 3}$ and ${p > \frac{d}{d-1}}$, although we will not focus on this application here.)

The condition ${p > \frac{d}{d-1}}$ can be seen to be necessary as follows. Take ${f}$ to be any fixed bump function. A brief calculation then shows that ${M_S f(x)}$ decays like ${|x|^{1-d}}$ as ${|x| \rightarrow \infty}$, and hence ${M_S f}$ does not lie in ${L^p({\bf R}^d)}$ unless ${p > \frac{d}{d-1}}$. By taking ${f}$ to be a rescaled bump function supported on a small ball, one can show that the condition ${p > \frac{d}{d-1}}$ is necessary even if we replace ${{\bf R}^d}$ with a compact region (and similarly restrict the radius parameter ${r}$ to be bounded). The condition ${d \geq 3}$ however is not quite necessary; the result is also true when ${d=2}$, but this turned out to be a more difficult result, obtained first by Bourgain, with a simplified proof (based on the local smoothing properties of the wave equation) later given by Muckenhaupt-Seeger-Sogge.

The Hardy-Littlewood maximal operator ${Mf}$, which involves averaging over balls, is clearly related to the spherical maximal operator, which averages over spheres. Indeed, by using polar co-ordinates, one easily verifies the pointwise inequality

$\displaystyle Mf(x) \leq M_S f(x)$

for any (continuous) ${f}$, which intuitively reflects the fact that one can think of a ball as an average of spheres. Thus, we see that the spherical maximal inequality (3) implies the Hardy-Littlewood maximal inequality (2) with the same constant ${C_{p,d}}$. (This implication is initially only valid for continuous functions, but one can then extend the inequality (2) to the rest of ${L^p({\bf R}^d)}$ by a standard limiting argument.)

At first glance, this observation does not immediately establish Theorem 1 for two reasons. Firstly, Stein’s spherical maximal theorem is restricted to the case when ${d \geq 3}$ and ${p > \frac{d}{d-1}}$; and secondly, the constant ${C_{d,p}}$ in that theorem still depends on dimension ${d}$. The first objection can be easily disposed of, for if ${p>1}$, then the hypotheses ${d \geq 3}$ and ${p > \frac{d}{d-1}}$ will automatically be satisfied for ${d}$ sufficiently large (depending on ${p}$); note that the case when ${d}$ is bounded (with a bound depending on ${p}$) is already handled by the classical maximal inequality (2).

We still have to deal with the second objection, namely that constant ${C_{d,p}}$ in (3) depends on ${d}$. However, here we can use the method of rotations to show that the constants ${C_{p,d}}$ can be taken to be non-increasing (and hence bounded) in ${d}$. The idea is to view high-dimensional spheres as an average of rotated low-dimensional spheres. We illustrate this with a demonstration that ${C_{d+1,p} \leq C_{d,p}}$, in the sense that any bound of the form

$\displaystyle \| M_S f \|_{L^p({\bf R}^d)} \leq A \|f\|_{L^p({\bf R}^d)} \ \ \ \ \ (4)$

for the ${d}$-dimensional spherical maximal function, implies the same bound

$\displaystyle \| M_S f \|_{L^p({\bf R}^{d+1})} \leq A \|f\|_{L^p({\bf R}^{d+1})} \ \ \ \ \ (5)$

for the ${d+1}$-dimensional spherical maximal function, with exactly the same constant ${A}$. For any direction ${\omega_0 \in S^d \subset {\bf R}^{d+1}}$, consider the averaging operators

$\displaystyle M_S^{\omega_0} f(x) := \sup_{r>0} A_r^{\omega_0} |f|(x)$

for any continuous ${f: {\bf R}^{d+1} \rightarrow {\bf C}}$, where

$\displaystyle A_r^{\omega_0} f(x) := \int_{S^{d-1}} f( x + r U_{\omega_0} \omega)\ d\sigma^{d-1}(\omega)$

where ${U_{\omega_0}}$ is some orthogonal transformation mapping the sphere ${S^{d-1}}$ to the sphere ${S^{d-1,\omega_0} := \{ \omega \in S^d: \omega \perp \omega_0\}}$; the exact choice of orthogonal transformation ${U_{\omega_0}}$ is irrelevant due to the rotation-invariance of surface measure ${d\sigma^{d-1}}$ on the sphere ${S^{d-1}}$. A simple application of Fubini’s theorem (after first rotating ${\omega_0}$ to be, say, the standard unit vector ${e_d}$) using (4) then shows that

$\displaystyle \| M_S^{\omega_0} f \|_{L^p({\bf R}^{d+1})} \leq A \|f\|_{L^p({\bf R}^{d+1})} \ \ \ \ \ (6)$

uniformly in ${\omega_0}$. On the other hand, by viewing the ${d}$-dimensional sphere ${S^d}$ as an average of the spheres ${S^{d-1,\omega_0}}$, we have the identity

$\displaystyle A_r f(x) = \int_{S^d} A_r^{\omega_0} f(x)\ d\sigma^d(\omega_0);$

indeed, one can deduce this from the uniqueness of Haar measure by noting that both the left-hand side and right-hand side are invariant means of ${f}$ on the sphere ${\{ y \in {\bf R}^{d+1}: |y-x|=r\}}$. This implies that

$\displaystyle M_S f(x) \leq \int_{S^d} M_S^{\omega_0} f(x)\ d\sigma^d(\omega_0)$

and thus by Minkowski’s inequality for integrals, we may deduce (5) from (6).

Remark 1 Unfortunately, the method of rotations does not work to show that the constant ${C_d}$ for the weak ${(1,1)}$ inequality (1) is independent of dimension, as the weak ${L^1}$ quasinorm ${\| \|_{L^{1,\infty}}}$ is not a genuine norm and does not obey the Minkowski inequality for integrals. Indeed, the question of whether ${C_d}$ in (1) can be taken to be independent of dimension remains open. The best known positive result is due to Stein and Strömberg, who showed that one can take ${C_d = Cd}$ for some absolute constant ${C}$, by comparing the Hardy-Littlewood maximal function with the heat kernel maximal function

$\displaystyle \sup_{t > 0} e^{t\Delta} |f|(x).$

The abstract semigroup maximal inequality of Dunford and Schwartz (discussed for instance in these lecture notes of mine) shows that the heat kernel maximal function is of weak-type ${(1,1)}$ with a constant of ${1}$, and this can be used, together with a comparison argument, to give the Stein-Strömberg bound. In the converse direction, it is a recent result of Aldaz that if one replaces the balls ${B(x,r)}$ with cubes, then the weak ${(1,1)}$ constant ${C_d}$ must go to infinity as ${d \rightarrow \infty}$.