### 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 . On the other hand, we have

For simplicity, we write

Then

By Cauchy-Schwarz inequality, we have

where and are associated square functions of .

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 and . Hence we get

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

We also define

and we denote it by . One can show that is well-defined for . This concept will be used in the proof of the following lemma.

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.