In this paper, we will introduce and study several types of Kakeya inequalities by the maximal functions in Hardy spaces in ℝⁿ, (n ≥ 2), and we could obtain several inequalities associated with the Kakeya inequalities. We will show that , when f(x) ∈ L^p(ℝⁿ) 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 ℝⁿ is n. The case for n ≥ 3 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):

(1)

where

is a 1 × δ tube centered at a ∈ ℝⁿ with the direction ξ ∈ Sⁿ⁻¹.

(2)

where the supremum is taken over all 1 × δ tubes T that contain x ∈ ℝⁿ. Formula (1) is Kakeya maximal function, and Formula (2) is Nikodym maximal function. When n = 2, in [3], Cordoba proved that for any ε > 0,

(3)

The Kakeya maximal function conjecture is formulated by Bourgain [4] that

(4)

holds for p ≥ n and n ∈ ℕ, and

(5)

holds for 1 < p ≤ n, q = (n − 1)p^′, and n ∈ ℕ. In 1983, Drury proved Formula (5) for p = (d + 1)/2, q = n + 1, in [5]. In 1991, Bourgain in [4] improved this result for each n ≥ 3 to some p(d) ∈ ((d + 1)/2, (d + 2)/2). By the interpolation theory, (see [6–8] and reference therein),

(6)

holds for p ≥ n and n ∈ ℕ.

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 δ = 1, and then

and

are classical maximal functions in Hardy spaces. And

(7)

for some constant C > 0.

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 δ^{−(n − 1)/p−ε} 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 δ = 1. In Theorem 7, the coefficient is the same as the factor δ^−ε in Formula (4) when . In Theorem 8, the coefficient is independent on δ when .

1.2. Notations

As usual, we use n to denote the dimension of ℝⁿ. suppf(x) is the support set of f(x). If x ∈ ℝⁿ: x = (x₁, x₂, ⋯, x_n), |x|_e denotes For α ∈ ℕⁿ: α = (α₁, α₂, ⋯, α_n), |α| denotes |α| = α₁ + α₂ + ⋯+α_n. We use ‖.‖_p to denote and O(ℝⁿ) to denote the n × n unit orthogonal matrix in ℝⁿ: O(ℝⁿ) = {A : A^TA = 1.A^T is the transposed matrix of A}.S(ℝⁿ) designates the space of C^∞ functions on ℝⁿ rapidly decreasing together with their derivatives. S_α,β(ℝⁿ) denotes and ε a positive fixed number (may be very small): ε > 0.

If X and Y are two quantities, X≲Y or Y ≳ X denotes that X ≤ CY for some absolute constant C > 0. More generally, given some parameters a₁, ⋯, a_k, we use or to denote the statement that for some constant which can depend on the parameter a₁, ⋯, a_k. We use X ~ Y to denote the statement X≲Y≲X, and similarly, denotes .

2. Preliminaries

For t, ξ ∈ ℝⁿ, f ∈ S(ℝⁿ), the Fourier transform of f is given by

(8)

and thus

where

and ∨ is the inversion of Fourier transform. For g ∈ S(ℝⁿ), g_I(x) designates

(9)

Let u = (u₁, u₂, ⋯, u_n) = xA⁻¹ = (x₁, x₂, ⋯, x_n)A⁻¹ where A is a variable (not fixed) matrix with A ∈ O(ℝⁿ), and then g_AI(x) is given by

(10)

If A is a variable (not fixed) matrix with A ∈ O(ℝⁿ), let

(11)

and thus

(12)

In this paper, let φ ∈ S(ℝⁿ) always to be a fixed radial function satisfying the following:

(13)

M_Yf(x) and

are given by

And nontangential maximal functions (f∗Y)_∇(x) are defined as usual: (f∗Y)_∇(x) = sup_{|x − y|≤t}|(f∗Y_t)(y)|. The even larger tangential variant

depending on a parameter N is given by

(14)

Let f to be a distribution, and Hardy spaces H^p(ℝⁿ) are (c.f. [9]): , for 0 < p < ∞ with appropriate α and β depending on p. It is known that H^p = L^p for p > 1:

In this paper, the Kakeya type maximal function

is given by

(15)

where

For some fixed t > 0,

can be defined by

(16)

Lemma 1 (see [9].)For any ψ ∈ S(ℝⁿ), 1 < p < ∞, f ∈ L^p(ℝⁿ), and N > n/p, we could obtain

(17)

Lemma 2 (see[10]). Let 0 < C₀ < ∞ and 0 < r < ∞. Then, there exist constants C₁ and C₂ (that depend only on n, C₀, and r) such that for all t > 0 and for all C¹(ℝⁿ) functions u on ℝⁿ whose Fourier transform is supported in the ball |ξ|_e ≤ C₀t and that satisfies for some B > 0, we have the estimation

(18)

where M denotes the Hardy-Littlewood maximal operator. (The constants C₁ and C₂ are independent of B and u.)

Lemma 3 (Phragmen-Lindelöf Lemma). Let F be analytic in the open strip S = {z ∈ ℂ : 0 < Rez < 1}, continuous, and bounded on its closure, such that |F(z)| ≤ C₀ when Rez = 0 and |F(z)| ≤ C₁ when Rez = 1. Then, when Rez = θ for any 0 < θ < 1.

3. The Case when 2 ≤ δ^−ε

When 1 ≤ δ^−ε ≤ 2, the case that δ~_ε1 is trival for the Kakeya type inequalities; thus, we only want to discuss the case when 0 < δ ≪ 1. In the following of this paper, we will discuss under the assumption that 2 ≤ δ^−ε.

3.1. Decomposition of the Phase Space

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

(19)

Then, we will give a decomposition of the region {ξ ∈ ℝⁿ : |ξ|_e ≤ 1} ⋃ {ξ ∈ ℝⁿ : 1 ≤ |ξ|_e ≤ δ^−4ε} into a collection of smaller ones:

(20)

Let the functions

for k ∈ ℤ, k ≥ 0 to be defined as

(21)

Then the functions

for k ∈ ℤ, k ≥ 0 are given by

(22)

Thus, it is clear that

and

Also, we could deduce that

(23)

In the same way, we could define the functions

and

for k ∈ ℤ, k ≥ 0 as

(24)

Then, we could deduce that

and

for k ≥ 1 hold. Thus, we could have

(25)

Notice that δ^{1+(k + 3)ε}|ξ|_e ≤ 1 for

. Then, we could obtain

(26)

We set

, (for 0 ≤ k, k ∈ ℤ) as

(27)

It is easy to see that

. ∃s ∈ ℕ, such that

(28)

We set

(k = 0, 1, 2, ⋯, s) as

(29)

Notice that 2^{−(k + 1)}δ|ξ|_e ≤ 1 holds, when

. Thus, we could obtain

(30)

Thus,

(31)

Thus, we could write

and

as radial, and A is a variable (not fixed) matrix with A ∈ O(ℝⁿ)) as

(32)

(33)

where 2^s ~ δ^−4ε.

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 N ≥ 0, N ∈ ℝ, k ∈ ℕ, k ≥ 2, and Y ∈ S_α,β(ℝⁿ) with appropriate α, β depending on ε, n, N, we have

(34)

Proof. First, we will prove that for l ∈ ℝ, l ≥ 0, k ∈ ℕ, and k ≥ 2, the following inequality holds:

(35)

Notice that the following inequality holds for 0 < δ < 1, for any m ∈ ℤ, m ≥ 0:

(36)

Thus, by the formula of integration by parts, we could deduce the following for any m ∈ ℤ, m ≥ 0:

(37)

Make a variable substitution:

(38)

We could write Formula (37) as

(39)

where Δ_ξ′ is the Laplace operator:

. We could also deduce that

(40)

Thus, , and

(41)

When , k ≥ 2, and δ^−ε ≥ 2, we could deduce that

(42)

That is,

(43)

Then, we could deduce that

(44)

Thus, similar to Formula (44), we could obtain

(45)

where k ≥ 2, m ∈ ℤ, m ≥ 0. By Lemma 3 and Formula (44), we could deduce Formula (35). Thus, we could obtain the following inequality for N ≥ 0, N ∈ ℝ

(46)

By Lemma 3 and Formula (45), for l ∈ ℝ, l ≥ 0, k ∈ ℕ, and k ≥ 2, the following inequality holds:

(47)

Then, we could obtain the following inequality for N ≥ 0, N ∈ ℝ, k ∈ ℕ, and k ≥ 2,

(48)

By Formulas (46) and (48), we could deduce that for N ≥ 0, N ∈ ℝ, k ∈ ℕ, and k ≥ 2, the following Formulas (49) and (50) hold:

(49)

(50)

Then, we could obtain the Lemma 4 directly from Formulas (49) and (50). This proves the Lemma.

Lemma 5. For N ≥ 0, N ∈ ℝ, k ∈ ℕ, and 0 ≤ k ≤ s where 2^s ~ δ^−4ε, Y ∈ S_α,β(ℝⁿ), with appropriate α, β depending on ε, n, N, the following two inequalities hold:

(51)

(52)

Proof. Notice that 2^s ~ δ^−4ε, thus, for any k ∈ {0, 1, ⋯, s} and N ≥ 0, N ∈ ℝ, we have

(53)

Notice that 2^{−(k + 1)}δ|ξ|_e ≤ 1 holds, when . Thus, we could obtain

(54)

Then, we could write as

(55)

It is clear that the following Formulas (56)–(60) hold:

(56)

(57)

(58)

(59)

(60)

By Young inequality, we could have

(61)

By the formula of integration by parts, we could deduce the following for any m ∈ ℕ:

(62)

By Young Inequality, Formulas (62) and (56)–(60), we could obtain

(63)

By Lemma 3 and Formulas (61) and (63), we could deduce the following Formula (64).

(64)

By Formulas (61) and (64), the following two inequalities hold for N ≥ 0, k ∈ {0, 1, ⋯, s}:

(65)

(66)

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 N ≥ 0, N ∈ ℝ, k ∈ ℕ, and k ≥ 2, we have

(67)

where A is a variable (not fixed) matrix with A ∈ O(ℝⁿ). For N ≥ 0, N ∈ ℝ, k ∈ ℕ, and 0 ≤ k ≤ s where 2^s ~ δ^−4ε, we have Formulas (37) and (68)

(68)

(69)

where A is a variable (not fixed) matrix with A ∈ O(ℝⁿ).

3.3. Main Results

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

Proposition 6. For p > 1 with appropriate α, β depending on ε, n, p, we have

(70)

(71)

where

Proof. Notice that f ∈ L^p(ℝⁿ) is a distribution. By Formula (33), we could obtain

(72)

where 2^s ~ δ^−4ε. Lemma 1 and Formulas (67), (69), and (72) yield to

(73)

Similar to Formula (72), we could also obtain

(74)

Notice that 2^s ~ δ^−4ε; thus, Lemma 1 and Formulas (67), (68), and (74) yield to

(75)

Let N be N = (n/p) + ε. From Formulas (73) and (75), we could prove the Proposition 6.

Theorem 7. For ∞>p > r > 1, 0 < t ≤ δ^−ε, f(x) ∈ L^p(ℝⁿ), and . Then, with appropriate α, β depending on ε, n, r, we could obtain

(76)

Proof. By Formula (33), we could write as

(77)

where 2^s ~ δ^−4ε.

By Holder inequality, and are both bounded functions for any x ∈ ℝ, δ > 0, k ∈ ℕ, and t > 0. Thus, for some B > 0, we could have

(78)

It is also clear that , and . We could also deduce that . Thus, by Lemma 2, we could obtain

(79)

(80)

Notice that 2^s ~ δ^−4ε, by Formulas (67), (69), (77), (79), and (80), we could deduce that for ∞>p > r > 1, 0 < t ≤ 1

(81)

When 1 < t ≤ δ^−ε, notice that

(82)

(83)

hold. From Formulas (67), (69), (77), (79), (80), (82), and (83), we could deduce that for ∞>p > r > 1, 1 < t ≤ δ^−ε

(84)

This proves the Theorem.

Theorem 8. For ∞>p > 1, 0 < t ≤ δ^−ε, f(x) ∈ L^p(ℝⁿ), and , then with appropriate α, β depending on ε, n, p, we could obtain

(85)

Proof. By Formula (33), we could write as the following:

(86)

Notice that and for 0 ≤ k ≤ s, k ∈ ℤ; thus, we could deduce that . From Formula (86), we could obtain

(87)

By Formula (67) and Lemma 1, we could have

(88)

Let N be N = (n/p) + ε, from Formulas (86)–(88), and we could prove the Theorem 8.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Open Research

Data Availability

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

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Kakeya Inequalities by Maximal Functions in Hardy Spaces

Abstract

1. Introduction

1.1. Main Result

1.2. Notations

2. Preliminaries

3. The Case when 2 ≤ δ^−ε

3.1. Decomposition of the Phase Space

3.2. Two Lemmas

3.3. Main Results

Conflicts of Interest

Open Research

Data Availability

References

References

Information

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Kakeya Inequalities by Maximal Functions in Hardy Spaces

Abstract

1. Introduction

1.1. Main Result

1.2. Notations

2. Preliminaries

3. The Case when 2 ≤ δ−ε

3.1. Decomposition of the Phase Space

3.2. Two Lemmas

3.3. Main Results

Conflicts of Interest

Open Research

Data Availability

References

References

Related

Information

3. The Case when 2 ≤ δ^−ε