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If is a locally integrable function, we define the Hardy-Littlewood maximal function
by the formula
where is the ball of radius
centred at
, and
denotes the measure of a set
. The Hardy-Littlewood maximal inequality asserts that
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.
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 :
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, implies the same bound
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
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
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
.
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