Kakeya Inequalities by Maximal Functions in Hardy Spaces

Abstract

In this paper, we will introduce and study several types of Kakeya inequalities by the maximal functions in Hardy spaces in , , and we could obtain several inequalities associated with the Kakeya inequalities. We will show that , when and .

1. Introduction

In 1917, Kakeya [1] proposed a problem to determine the minimal area needed to continuously rotate a unit line segment in the plane by 180 degrees. In 1928, Besicovithch [2] proved the measure of such sets could be arbitrary small. Such sets are called Besicovitch sets or Kakeya sets. The Kakeya conjectures state that the Hausdorff dimension of any Besicovitch sets in is . The case for is still an open problem. The so-called maximal Kakeya conjecture (or maximal Nikodym conjecture) is actually a stronger one that involves the following Kakeya maximal function (or Nikodym maximal function): where is a tube centered at with the direction . where the supremum is taken over all tubes that contain . Formula (1) is Kakeya maximal function, and Formula (2) is Nikodym maximal function. When , in [3], Cordoba proved that for any ,

The Kakeya maximal function conjecture is formulated by Bourgain [4] that holds for and , and holds for , , and . In 1983, Drury proved Formula (5) for , , in [5]. In 1991, Bourgain in [4] improved this result for each to some . By the interpolation theory, (see [6–8] and reference therein), holds for and .

1.1. Main Result

Inspired by the Formulas (1), (2), and (4)–(6), we will consider maximal functions like and in this paper. Notice that the classical case is , and then and are classical maximal functions in Hardy spaces. And for some constant .

We will obtain several inequalities in Proposition 6, Theorem 7, and Theorem 8. In Proposition 6, though the coefficient in Formula (71) is not better than the factor in Formula (6), we use a way different to [3, 4] and [5]. And we could obtain Formula (70) which is different to the classical case . In Theorem 7, the coefficient is the same as the factor in Formula (4) when . In Theorem 8, the coefficient is independent on when .

1.2. Notations

As usual, we use to denote the dimension of . is the support set of . If : , denotes For : , denotes We use to denote and to denote the unit orthogonal matrix in : designates the space of functions on rapidly decreasing together with their derivatives. denotes and a positive fixed number (may be very small): .

If and are two quantities, or denotes that for some absolute constant . More generally, given some parameters , we use or to denote the statement that for some constant which can depend on the parameter . We use to denote the statement , and similarly, denotes .

2. Preliminaries

For , , the Fourier transform of is given by and thus where and is the inversion of Fourier transform. For , designates

Let where is a variable (not fixed) matrix with , and then is given by

If is a variable (not fixed) matrix with , let and thus

In this paper, let always to be a fixed radial function satisfying the following:

and are given by And nontangential maximal functions are defined as usual: The even larger tangential variant depending on a parameter is given by

Let to be a distribution, and Hardy spaces are (c.f. [9]): , for with appropriate and depending on . It is known that for :

In this paper, the Kakeya type maximal function is given by where For some fixed , can be defined by

Lemma 1 (see [9]). For any , , , and , we could obtain

Lemma 2 (see[10]). Let and . Then, there exist constants and (that depend only on , , and ) such that for all and for all functions on whose Fourier transform is supported in the ball and that satisfies for some , we have the estimation where denotes the Hardy-Littlewood maximal operator. (The constants and are independent of and .)

Lemma 3 (Phragmen-Lindelöf Lemma). Let be analytic in the open strip , continuous, and bounded on its closure, such that when and when . Then, when for any .

3. The Case when

When , the case that is trival for the Kakeya type inequalities; thus, we only want to discuss the case when . In the following of this paper, we will discuss under the assumption that .

3.1. Decomposition of the Phase Space

In this section, we will decompose into a collection of regions:

Then, we will give a decomposition of the region into a collection of smaller ones:

Let the functions for to be defined as

Then the functions for are given by

Thus, it is clear that and Also, we could deduce that

In the same way, we could define the functions and for as

Then, we could deduce that and for hold. Thus, we could have

Notice that for . Then, we could obtain

We set , () as

It is easy to see that . , such that

We set () as

Notice that holds, when . Thus, we could obtain

Thus,

Thus, we could write and as radial, and is a variable (not fixed) matrix with ) as

where .

3.2. Two Lemmas

In this section, we will estimate the integrals (in Lemmas 4 and 5) associated with and given in Formulas (32) and (33).

Lemma 4. For , , , and with appropriate depending on , we have

Proof. First, we will prove that for , , , and , the following inequality holds: Notice that the following inequality holds for , for any : Thus, by the formula of integration by parts, we could deduce the following for any : Make a variable substitution: We could write Formula (37) as where is the Laplace operator: . We could also deduce that Thus, , and When , , and , we could deduce that That is, Then, we could deduce that Thus, similar to Formula (44), we could obtain where , . By Lemma 3 and Formula (44), we could deduce Formula (35). Thus, we could obtain the following inequality for By Lemma 3 and Formula (45), for , , , and , the following inequality holds: Then, we could obtain the following inequality for , , and , By Formulas (46) and (48), we could deduce that for , , and , the following Formulas (49) and (50) hold: Then, we could obtain the Lemma 4 directly from Formulas (49) and (50). This proves the Lemma.

Lemma 5. For , , and where , , with appropriate depending on , the following two inequalities hold:

Proof. Notice that , thus, for any and , we have Notice that holds, when . Thus, we could obtain Then, we could write as It is clear that the following Formulas (56)–(60) hold: By Young inequality, we could have By the formula of integration by parts, we could deduce the following for any : By Young Inequality, Formulas (62) and (56)–(60), we could obtain By Lemma 3 and Formulas (61) and (63), we could deduce the following Formula (64). By Formulas (61) and (64), the following two inequalities hold for , : From Formulas (53), (65), and (66), we could obtain the Formula (51) and (52) together. This proves the Lemma.

From Lemmas 4 and 5, we could obtain the following inequalities (67)–(69). For , , and , we have where is a variable (not fixed) matrix with . For , , and where , we have Formulas (37) and (68) where is a variable (not fixed) matrix with .

3.3. Main Results

From Formulas (67)–(69), we will obtain our main results in this section:

Proposition 6. For with appropriate , depending on , we have where

Proof. Notice that is a distribution. By Formula (33), we could obtain where . Lemma 1 and Formulas (67), (69), and (72) yield to Similar to Formula (72), we could also obtain Notice that ; thus, Lemma 1 and Formulas (67), (68), and (74) yield to Let be . From Formulas (73) and (75), we could prove the Proposition 6.

Theorem 7. For , , , and . Then, with appropriate , depending on , we could obtain

Proof. By Formula (33), we could write as where .
By Holder inequality, and are both bounded functions for any , , , and . Thus, for some , we could have It is also clear that , and . We could also deduce that . Thus, by Lemma 2, we could obtain Notice that , by Formulas (67), (69), (77), (79), and (80), we could deduce that for , When , notice that hold. From Formulas (67), (69), (77), (79), (80), (82), and (83), we could deduce that for , This proves the Theorem.

Theorem 8. For , , , and , then with appropriate , depending on , we could obtain

Proof. By Formula (33), we could write as the following: Notice that and for ; thus, we could deduce that . From Formula (86), we could obtain By Formula (67) and Lemma 1, we could have Let be , from Formulas (86)–(88), and we could prove the Theorem 8.

Data Availability

There is no data in my paper; thus, the date is available.

Conflicts of Interest

The author declares that there are no conflicts of interest.