Uniform improving estimates of damped plane Radon transforms in Lebesgue and Lorentz spaces are studied under mild assumptions on the rotational curvature. The results generalize previously known estimates. Also, they extend sharp estimates known for convolution operators with affine arclength measures to the semitranslation-invariant case.

1. Introduction

Let 𝒟 be a domain in ℝ² and let

()

For x₁ ∈ 𝒟^* and y ∈ 𝒟_*, we write

()

To avoid technical difficulties, we assume that, for x₁ ∈ 𝒟^* and y ∈ 𝒟_*,

and 𝒟^y are finite intervals throughout the paper. For a C² function φ and a measurable function ω defined on 𝒟, we consider the damped Radon transform ℛ_φ,ω defined by

()

for

. Mapping properties of such operators in various function spaces have been studied by many authors [1–9]. Sharper estimates are available in translation-invariant cases where φ(x₁, y) = ϕ(x₁ − y) with a C² function ϕ defined on an interval [10, 11] and it is widely known that the so-called affine arclength measure introduced by Drury [12] is better suited in obtaining degeneracy independent results in many interesting cases. Analogous quantity in nontranslation-invariant situation is rotational curvature, which is given by

in this setting. In this paper, we are interested in uniform optimal improving properties in Lebesgue spaces and Lorentz spaces. The results will generalize known estimates for damped Radon transform and convolution operators with affine arclength measure on plane curves.

Before we state the results, we introduce certain conditions on functions defined on intervals. For an interval J₁ in ℝ, a locally integrable function Φ : J₁ → ℝ⁺, and a positive real number A, we let

()

An interesting subclass of ℰ₁(2A) is the collection ℰ₂(A), introduced in [13], of functions Φ : J → ℝ⁺ such that

(1)
Φ is monotone,
(2)
whenever s₁ < s₂ and [s₁, s₂] ⊂ J,
()

In connection with the problems related to convolution operators with affine arclength measure on curves in the plane, the author of [10] proved the following.

Theorem 1. Let J be an open interval in ℝ, and let ϕ : J → ℝ be a C² function such that ϕ^′′ ≥ 0. Let ω : J → ℝ be a nonnegative measurable function. Suppose that there exists a positive constant A such that ω ∈ 𝔊(ϕ^′′, A); that is,

()

holds whenever s₁ < s₂ and [s₁, s₂] ⊂ J. Let 𝒮_ϕ be the operator given by

()

for

. Then, there exists a constant C that depends only on A such that

()

holds uniformly in

Regarding the endpoint Lorentz space estimates, the following result due to Oberlin is available.

Theorem 2 (Oberlin [11]). Let ϕ be a C² function on an interval J such that ϕ^′′ > 0 on J and ϕ^′′ ∈ ℰ₂(A). Then, 𝒮_ϕ defined in (7) maps L^3/2,3(ℝ²) boundedly to L³(ℝ²) with the operator norm depending only on A.

In this paper, the author generalizes the aforementioned theorems to damped Radon transforms where the condition on the affine arclength measure is replaced by that on the rotational curvature. This paper is organized as follows: in Section 2, uniform estimate in Lebesgue spaces is studied, and in Section 3, endpoint Lorentz space estimate will be given based on an approach similar to Oberlin’s approach [11, 14].

2. Uniform Estimates on the Plane

Theorem 3. Let φ be a C² function on 𝒟 such that , and let ω be a nonnegative measurable function on 𝒟. Suppose that there exists a positive constant A such that, for each x₁ ∈ 𝒟^*, ; that is,

()

holds whenever y₁ < y₂ and

. Let ℛ_φ,ω be the operator given by (3). Then, there exists a constant C that depends only on A such that

()

holds uniformly in

Proof of Theorem 3. Our proof is based on the method introduced by Drury and Guo [15], which was later refined by Oberlin [16] and the author of [10]. We have

()

where for y₁, y₂, y₃ ∈ 𝒟_* and suitable functions g₁, g₂, g₃ defined on ℝ,

()

with

. As in the proof of Theorem 2.1 in [10], one can show that the estimate

()

holds uniformly in g₁, g₂, g₃, y₁, y₂, and y₃. Combining this with Proposition 2.2 in the work by Christ [17] finishes the proof.

Remark 4. The special case in which provides a uniform estimate for the damped plane Radon transform. We write

()

for

Corollary 5. Let φ : 𝒟 → ℝ be a C² function such that . Suppose that there exists a constant A such that, for each x₁ ∈ 𝒟^*, ; that is,

()

holds whenever y₁ < y₂ and

. Let ℛ_φ be the operator given by (14). Then, there exists a constant C that depends only on A such that

()

holds uniformly in

Remark 6. A duality argument shows the following.

Corollary 7. Let φ : 𝒟 → ℝ be a C² function such that . Suppose that there exists a constant A such that, for each y₁ ∈ 𝒟_*, ; that is,

()

holds whenever x₁ < x₂ and

. Let ℛ_φ be the operator given by (14). Then, there exists a constant C that depends only on A such that

()

holds uniformly in

3. Endpoint Lorentz Estimates

Under somewhat stronger condition, estimates in Section 2 can be improved. Namely, we have the following.

Theorem 8. Let φ : 𝒟 → ℝ be a C² function such that . Suppose that there exists a constant A such that, for each x₁ ∈ 𝒟^*, ; that is,

()

holds whenever y₁ < y₂ and

. Let ℛ_φ be the operator given by (14). Then, there exists a constant C that depends only on A such that

()

holds uniformly in

Proof of Theorem 8. To ease our notation, we let . For a measurable subset E of either ℝ or ℝ², we denote the Lebesgue measure and the characteristic function of E by |E| and 1_E, respectively.

By a well-known interpolation argument as in [2, 18], it suffices to establish the estimate

()

for all measurable subsets E₁, E₂, and E₃ of ℝ². We have

()

where

()

By Schwarz inequality, it suffices to get an estimate

()

uniformly in x₂, y₁, and E. By translation invariance of

in x₂ variable, it is enough to establish

()

uniformly in y₁ and E, where

()

Notice that the map Γ : (x₁, y) ↦ (y, φ(x₁, y) − φ(x₁, y₁)) is one-to-one and has the absolute value of Jacobian determinant

for a given y₁ ∈ 𝒟_*.

3.1. Estimate for ℬ₁

We follow an approach by Oberlin [14]. Letting

()

we have

()

Here, for z > y₁, we denoted by F_z the set F∩[z, ∞). On the other hand, applying Hölder’s inequality as in [14], we get

()

Combined with the monotonicity of ω(x₁, ·), we obtain

()

An integration in x₁ provides (25).

3.2. Estimate forℬ₂

For fixed x₁ and y₁, we let

()

where C(1) is the constant that appears in Lemma 2.2 in [11], which implies

()

Since ω(x₁, ·) is nondecreasing, we see

()

Note that the second inequality follows from a simple modification of Lemma 2.1 in [11]. This finishes the proof.

Remark 9. A duality argument shows the following.

Corollary 10. Let φ : 𝒟 → ℝ be a C² function such that . Suppose that there exists a constant A such that, for each y ∈ 𝒟_*, ; that is,

()

holds whenever x₁ < x₂ and [x₁, x₂] ⊂ 𝒟^y. Let ℛ_φ be the operator given by (14). Then, there exists a constant C that depends only on A such that

()

holds uniformly in

Remark 11. As is well known, if ℛ_φ maps boundedly from L^p,u(ℝ²) to L^q,v(ℝ²), then (1/p, 1/q) belongs to the convex hull of {(0,0), (1,1), (2/3,1/3)}, and uniform estimates are possible only if (1/p, 1/q) = (2/3,1/3). In the latter case, 3/2 ≤ v ≤ u ≤ 3 is necessary, implying the sharpness of the results. We refer interested readers to [2, 19].

Acknowledgment

This paper was completed with Ajou University Research Fellowship of 2011.

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Uniform Estimates for Damped Radon Transform on the Plane

Abstract

1. Introduction

2. Uniform Estimates on the Plane

3. Endpoint Lorentz Estimates

3.1. Estimate for ℬ₁

3.2. Estimate forℬ₂

Acknowledgment

References

References

Information

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Uniform Estimates for Damped Radon Transform on the Plane

Abstract

1. Introduction

2. Uniform Estimates on the Plane

3. Endpoint Lorentz Estimates

3.1. Estimate for ℬ1

3.2. Estimate forℬ2

Acknowledgment

References

References

Related

Information

3.1. Estimate for ℬ₁

3.2. Estimate forℬ₂