If is a locally integrable function, we define the Hardy-Littlewood maximal function by the formula
for all , all , and some constant depending only on . By a standard density argument, this implies in particular that we have the Lebesgue differentiation theorem
for all and almost every . 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 ) we see that
for all and , and some constant depending on and .
The exact dependence of on and 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 for some absolute constant . Inserting this into the Marcinkiewicz theorem, one obtains a constant of the form for some (and taking bounded away from infinity, for simplicity). The dependence on is about right, but the dependence on should not be exponential.
The argument is based on an earlier bound of Stein from 1976 on the spherical maximal function
where are the spherical averaging operators
and is normalised surface measure on the sphere . Because this is an uncountable supremum, and the averaging operators do not have good continuity properties in , it is not a priori obvious that is even a measurable function for, say, locally integrable ; but we can avoid this technical issue, at least initially, by restricting attention to continuous functions . The Stein maximal theorem for the spherical maximal function then asserts that if and , then we have
for all (continuous) . We will sketch a proof of this theorem below the fold. (Among other things, one can use this bound to show the pointwise convergence of the spherical averages for any when and , although we will not focus on this application here.)
The condition can be seen to be necessary as follows. Take to be any fixed bump function. A brief calculation then shows that decays like as , and hence does not lie in unless . By taking to be a rescaled bump function supported on a small ball, one can show that the condition is necessary even if we replace with a compact region (and similarly restrict the radius parameter to be bounded). The condition however is not quite necessary; the result is also true when , 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 , 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
for any (continuous) , 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 . (This implication is initially only valid for continuous functions, but one can then extend the inequality (2) to the rest of 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 and ; and secondly, the constant in that theorem still depends on dimension . The first objection can be easily disposed of, for if , then the hypotheses and will automatically be satisfied for sufficiently large (depending on ); note that the case when is bounded (with a bound depending on ) is already handled by the classical maximal inequality (2).
We still have to deal with the second objection, namely that constant in (3) depends on . However, here we can use the method of rotations to show that the constants can be taken to be non-increasing (and hence bounded) in . The idea is to view high-dimensional spheres as an average of rotated low-dimensional spheres. We illustrate this with a demonstration that , in the sense that any bound of the form
for the -dimensional spherical maximal function, with exactly the same constant . For any direction , consider the averaging operators
for any continuous , where
where is some orthogonal transformation mapping the sphere to the sphere ; the exact choice of orthogonal transformation is irrelevant due to the rotation-invariance of surface measure on the sphere . A simple application of Fubini’s theorem (after first rotating to be, say, the standard unit vector ) using (4) then shows that
uniformly in . On the other hand, by viewing the -dimensional sphere as an average of the spheres , we have the identity
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 on the sphere . This implies that
Remark 1 Unfortunately, the method of rotations does not work to show that the constant for the weak inequality (1) is independent of dimension, as the weak quasinorm is not a genuine norm and does not obey the Minkowski inequality for integrals. Indeed, the question of whether 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 for some absolute constant , by comparing the Hardy-Littlewood maximal function with the heat kernel maximal function
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 with a constant of , 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 with cubes, then the weak constant must go to infinity as .
— 1. Proof of spherical maximal inequality —
We now sketch the proof of Stein’s spherical maximal inequality (3) for , , and continuous. To motivate the argument, let us first establish the simpler estimate
where is the spherical maximal function restricted to unit scales:
For the rest of these notes, we suppress the dependence of constants on and , using as short-hand for .
for all continuous , as the original claim follows by replacing with . Also, since the bound is trivially true for , and we crucially have in three and higher dimensions, we can restrict attention to the regime .
We establish this bound using a Littlewood-Paley decomposition
where ranges over dyadic numbers , , and is a smooth Fourier projection to frequencies ; a bit more formally, we have
where is a bump function supported on the annulus such that for all non-zero . Actually, for the purposes of proving (7), it is more convenient to use the decomposition
for all and some depending only on .
To prove the low-frequency bound (8), observe that is a convolution operator with a Schwartz function, and from this and the radius restriction we see that is a convolution operator with a Schwartz function of uniformly bounded norms. From this we obtain the pointwise bound
Now we turn to the more interesting high-frequency bound (9). Here, is a convolution operator with an approximation to the identity at scale , and so is a convolution operator with a function of magnitude concentrated on an annulus of thickness around the sphere of radius . This can be used to give the pointwise bound
which by (2) gives the bound
for any . This is not directly strong enough to prove (9), due to the “loss of one derivative” as manifested by the factor . On the other hand, this bound (12) holds for all , and not just in the range .
To counterbalance this loss of one derivative, we turn to estimates. A standard stationary phase computation (or Bessel function computation) shows that is a Fourier multiplier whose symbol decays like . As such, Plancherel’s theorem yields the bound
uniformly in . But we still have to take the supremum over . This is an uncountable supremum, so one cannot just apply a union bound argument. However, from the uncertainty principle, we expect to be “blurred out” at spatial scale , which suggests that the averages do not vary much when is restricted to an interval of size . Heuristically, this then suggests that
Estimating the discrete supremum on the right-hand side somewhat crudely by the square-function,
One can make this heuristic precise using the one-dimensional Sobolev embedding inequality adapted to scale , namely that
To prove this inequality, one starts with the local one-dimensional Sobolev inequality
rescales this inequality to the scale , and then covers the interval by boundedly overlapping intervals of length .
A routine computation shows that
(which formalises the heuristic that is roughly constant at -scales ), and this soon leads to a rigorous proof of (13).
Now we control the full maximal function . It suffices to show that
where ranges over dyadic numbers.
For any fixed , the natural spatial scale is , and the natural frequency scale is thus . We therefore split
for each and some depending only on and , similarly to before.
A rescaled version of the derivation of (10) gives
for all . Meanwhile, at the level, we have
which implies by rescaled Sobolev embedding that
In fact, by writing , where is a slight widening of , we have
square summing this (and bounding a supremum by a square function) and using Plancherel we obtain