1. Introduction
Maximal function plays a central role in several places in analysis. As an example, we can prove the celebrated Lebesgue differentiation theorem by using Hardy-Littlewood maximal function. Another example for application of maximal function is the nontangential behavior of Poisson integral defined on the half-plane. From these examples, maximal function helps us to understand a pointwise behavior of certain family of functions. See Stein’s monograph (1970) in Chapters 1, 3 for details.
It is well known from classical theory on PDEs that
solves wave equation
Here denotes the normalized surface measure on the unit sphere
. Such formula is called Kirchoff’s formula. From this, it is natural to define
the spherical average of . Note that by Minkowski’s integral inequality,
is well-defined for
,
. As we mentioned before, to understand the pointwise behavior of spherical average of
, we define an associated maximal operator
by
We call the operator the spherical maximal operator. In 1976, Stein proved that the spherical maximal operator is bounded from
to itself when
and
. In the same paper, he proved that the
-boundedness of spherical maximal operator is failed when
and
. Consider
Then it is easy to see that and
but
. Hence it remains open when
. It was resolved by Bourgain (1986) in 1983 that
-boundedness of
holds for
when
. There are several extensions on this result. We also mentioned that
–
mapping property of spherical maximal function was proved by Schlag (1997, 1998) in
and Schlag-Sogge (1997) under more general settings in the connection with local smoothing estimate. We also mention the result of Mockenhaupt-Seeger-Sogge (1992), which is the first paper containing the connection between local smoothing estimate of wave equation and
-boundedness of circular maximal operator. A standard references on this topic are Stein-Wainger (1978) and the monograph of Stein (1993).
The goal of this note is to prove Stein’s original result.
Theorem 1.1. Let
and
. Then there exists a constant
such that
for all
.
Originally, Stein used some variant of -functions and its mapping property. Here we follow a quite modern strategy via Littlewood-Paley decomposition as presented in Rubio de Francia (1986). We write
Then we decompose
where is the usual Littlewood-Paley partition of unity. If we write
, we first prove that
and
for all and for all
. Here
Since and
, where
denotes the usual Hardy-Littlewood maximal operator, then the desired result follows by an interpolation and the
-boundedness of Hardy-Littlewood maximal operator.
The rest of this note is organized as follows. In Section 2, we list some facts we used in this note. We prove the main theorem in Section 3.
2. Preliminaries
In this section, we first give some results on Hardy-Littlewood maximal operator.
For , we denote by
the Hardy-Littlewood maximal function of
which is defined as
Theorem 2.1. We have
in other words,
For any
, there exists a constant
such that
for all
.
As an application of Theorem 2.1, we have the following Lebesgue differentiation theorem.
Theorem 2.2. Let
. Then
For the proof of Theorems 2.1 and 2.2, see Stein’s monograph (1970) for details.
Next, we will use some decay estimate on , which is defined by
It is easy to see that is smooth. Moreover, we have the following decay estimate.
Proposition 2.3. We have
for large
.
See Wolff’s textbook (2003) or Grafakos’s textbook (2014). A proof presented in Wolff (2003) does not involve Bessel function.
Finally, we will use the Marcinkiewicz interpolation theorem. See Grafakos’s textbook (2014) for the proof.
Theorem 2.4. Let
be a sublinear operator and
and let
be a sublinear operator defined on
and taking values in the space of measurable functions on
. Suppose that
and
for all
.
Then for
, define
Then there exists a constant
such that
for all
.
3. Proof of Theorem 1.1
This section is devoted to a proof of Theorem 1.1. Below we assume that and
. Set
and define
where satisfies
and
Note that if we write
then
We decompose our symbol as follows:
If we define
then
The following lemma will be used in several places.
Lemma 3.1. For any
, we have
where
denotes the standard Hardy-Littlewood maximal operator.
Proof. It suffices to show the case . We decompose
This proves the desired result.
From this lemma, we immediately get
since

Lemma 3.2. There exists a constant
such that
for all
and
.
Proof. For , we write
If we write
then
Since , it follows that
Hence by Lebesgue differentiation theorem, we get
From this it follows from the fundamental theorem of calculus that
(1)
for almost every

For simplicity, we write
Then
By Cauchy-Schwarz inequality, we have
where



We will show that
and
Indeed, by Plancherel’s theorem, we get
By Proposition 2.3, we have
Since is supported in the annulus
, it follows that
Here we used an elementary calculus
for


To estimate , we recall that
By Proposition 2.3, we have
Hence following the exactly same argument, one can get
Such estimate is natural since we have to consider one derivative gain. Therefore, we get
which completes the proof of Lemma 3.2.
Next, we show that each is of weak type
. To do this, we recall that for any finite Borel measure
, its Fourier transform is defined by
and we denote it by



Lemma 3.3. There exists a constant
such that
for all
and for all
.
Proof. We claim that
(2)
for . If this is true, then by Lemma 3.1, we have
Since maps
to
, it follows that
It remains to show (2). Recall that
where
If we write
then we have
By a standard argument, it suffices to estimate
We decompose our integral as follows. Write
and
One can easily show that
for any and
. Note also that if
, then
and so
since
. From this, we have
This proves the desired result.
We are ready to prove the main theorem.
Proof of Theorem 1.1. So far we proved that
and
for all . Hence by Marcinkiewicz interpolation theorem, we get
for all . Since
and
, it follows from the
-boundedness of Hardy-Littlewood maximal function that
Here we used the restriction to guarantee that
converges. This completes the proof of Theorem 1.1.