Uniform Estimates for Damped Radon Transform on the Plane

Abstract

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 𝒟^∗∶=𝑥₁∈ℝ∶𝑥₁,𝒟,𝑦∈𝒟forsome𝑦∈ℝ_∗∶=𝑦∈ℝ∶𝑥₁,𝑦∈𝒟forsome𝑥₁∈ℝ. (1) For 𝑥₁∈𝒟^∗ and 𝑦∈𝒟_∗, we write 𝒟_𝑥1∶=𝑧∈ℝ∶𝑥₁,𝒟,𝑧∈𝒟_𝑦∶=𝑧∈ℝ∶𝑧,𝑦∈𝒟. (2) To avoid technical difficulties, we assume that, for 𝑥₁∈𝒟^∗ and 𝑦∈𝒟_∗, 𝒟_𝑥1 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 ℛ_𝜑,𝜔𝑓𝑥₁,𝑥₂∶=_𝒟𝑥1𝑓𝑦,𝑥₂+𝜑𝑥₁,𝑦𝜔𝑥₁,𝑦𝑑𝑦 (3) for 𝑓∈𝐶^∞₀(ℝ²). 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 𝜑^′′₁₂(𝑥₁,𝑦)=𝜕²𝜑/𝜕𝑥₁𝜕𝑦 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 𝔊(Φ,𝐴)∶=𝜔∶𝐽₁⟶ℝ⁺∣𝜔𝑠₁𝜔𝑠₂≤𝐴𝑠₂−𝑠₁^𝑠2_𝑠1Φ(𝑠)𝑑𝑠whenever𝑠₁<𝑠₂and𝑠₁,𝑠₂⊂𝐽₁,ℰ₁(𝐴)∶=Φ∶𝐽⟶ℝ⁺∣Φ∈𝔊(Φ,𝐴). (4) 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, Φ𝑠₁Φ𝑠₂𝑠≤𝐴Φ¹+𝑠₂2. (5)

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, 𝜔𝑠₁^1/2𝜔𝑠₂^1/2≤𝐴𝑠₂−𝑠₁^𝑠2_𝑠1𝜙^′′(𝑣)𝑑𝑣 (6) holds whenever 𝑠₁<𝑠₂ and [𝑠₁,𝑠₂]⊂𝐽. Let 𝒮_𝜙 be the operator given by 𝒮_𝜙𝑓𝑥₁,𝑥₂=_𝐽𝑓𝑥₁−𝑠,𝑥₂−𝜙(𝑠)𝜔^1/3(𝑠)𝑑𝑠 (7) for 𝑓∈𝐶^∞₀(ℝ²). Then, there exists a constant 𝐶 that depends only on 𝐴 such that 𝒮_𝜙𝑓_𝐿3(ℝ2)≤𝐶𝑓_{𝐿3/2(ℝ2)} (8) holds uniformly in 𝑓∈𝐶^∞₀(ℝ²).

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 𝜑^′′₁₂>0, and let 𝜔 be a nonnegative measurable function on 𝒟. Suppose that there exists a positive constant 𝐴 such that, for each 𝑥₁∈𝒟^∗, 𝜔(𝑥₁,⋅)∈𝔊(𝜑^′′₁₂(𝑥₁,⋅),𝐴); that is, 𝜔𝑥₁,𝑦₁^1/2𝜔𝑥₁,𝑦₂^1/2≤𝐴𝑦₂−𝑦₁^𝑦2_𝑦1𝜑^′′₁₂𝑥₁,𝑧𝑑𝑧 (9) holds whenever 𝑦₁<𝑦₂ and [𝑦₁,𝑦₂]⊂𝒟_𝑥1. Let ℛ_𝜑,𝜔 be the operator given by (3). Then, there exists a constant 𝐶 that depends only on 𝐴 such that ℛ_𝜑,𝜔𝑓_𝐿3(ℝ2)≤𝐶𝑓_{𝐿3/2(ℝ2)} (10) holds uniformly in 𝑓∈𝐶^∞₀(ℝ²).

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 ℛ_𝜑,𝜔𝑓³₃=_ℝ𝒟∗³_{𝑗=1𝒟𝑥1}𝑓𝑦₁,𝑥₂+𝜑𝑥₁,𝑦_𝑗×𝜔^1/3𝑥₁,𝑦_𝑗𝑑𝑦_𝑗𝑑𝑥₁𝑑𝑥₂=_𝒟∗𝒢𝑓𝑦₁,⋅,𝑓𝑦₂,⋅,𝑓𝑦₃×𝑦,⋅₁,𝑦₂,𝑦₃𝑑𝑦₁𝑑𝑦₂𝑑𝑦₃, (11) where for 𝑦₁,𝑦₂,𝑦₃∈𝒟_∗ and suitable functions 𝑔₁, 𝑔₂, 𝑔₃ defined on ℝ,𝒢𝑔₁,𝑔₂,𝑔₃𝑦₁,𝑦₂,𝑦₃∶=_{ℝ𝒟𝑦123,𝑦,𝑦}³_𝑗=1𝑔_𝑗𝑥₂+𝜑𝑥₁,𝑦_𝑗𝜔^1/3𝑥₁,𝑦_𝑗𝑑𝑥₁𝑑𝑥₂ (12)with 𝒟^{𝑦1,𝑦2,𝑦3}∶=⋂³_𝑗=1𝒟^𝑦𝑗. As in the proof of Theorem 2.1 in [10], one can show that the estimate 𝒢𝑔₁,𝑔₂,𝑔₃𝑦₁,𝑦₂,𝑦₃≤𝐶𝑔_{1𝐿3/2(ℝ)}𝑔_{2𝐿3/2(ℝ)}𝑔_{3𝐿3/2(ℝ)}𝑦₁−𝑦₂𝑦₁−𝑦₃𝑦₂−𝑦₃^1/3 (13) 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 𝜔=𝜑^′′₁₂ provides a uniform estimate for the damped plane Radon transform. We write ℛ_𝜑𝑓𝑥₁,𝑥₂=_𝒟𝑥1𝑓𝑦,𝑥₂+𝜑𝑥₁,𝑦𝜑^′′₁₂𝑥₁,𝑥₂,𝑦^1/3𝑑𝑦 (14) for 𝑓∈𝐶^∞₀(ℝ²).

Corollary 5. Let 𝜑∶𝒟→ℝ be a 𝐶² function such that 𝜑^′′₁₂>0. Suppose that there exists a constant 𝐴 such that, for each 𝑥₁∈𝒟^∗, 𝜑^′′₁₂(𝑥₁,⋅)∈ℰ₁(𝐴); that is, 𝜑^′′₁₂𝑥₁,𝑦₁^1/2𝜑^′′₁₂𝑥₁,𝑦₂^1/2≤𝐴𝑦₂−𝑦₁^𝑦2_𝑦1𝜑^′′₁₂𝑥₁,𝑧𝑑𝑧 (15) holds whenever 𝑦₁<𝑦₂ and [𝑦₁,𝑦₂]⊂𝒟_𝑥1. Let ℛ_𝜑 be the operator given by (14). Then, there exists a constant 𝐶 that depends only on 𝐴 such that ℛ_𝜑𝑓_𝐿3(ℝ2)≤𝐶𝑓_{𝐿3/2(ℝ2)} (16) holds uniformly in 𝑓∈𝐶^∞₀(ℝ²).

Remark 6. A duality argument shows the following.

Corollary 7. Let 𝜑∶𝒟→ℝ be a 𝐶² function such that 𝜑^′′₁₂>0. Suppose that there exists a constant 𝐴 such that, for each 𝑦₁∈𝒟_∗, 𝜑^′′₁₂(⋅,𝑦₁)∈ℰ₁(𝐴); that is, 𝜑^′′₁₂𝑥₁,𝑦₁^1/2𝜑^′′₁₂𝑥₂,𝑦₁^1/2≤𝐴𝑥₂−𝑥₁^𝑥2_𝑥1𝜑^′′₁₂𝑧,𝑦₁𝑑𝑧 (17) holds whenever 𝑥₁<𝑥₂ and [𝑥₁,𝑥₂]⊂𝒟^𝑦1. Let ℛ_𝜑 be the operator given by (14). Then, there exists a constant 𝐶 that depends only on 𝐴 such that ℛ_𝜑𝑓_𝐿3(ℝ2)≤𝐶𝑓_{𝐿3/2(ℝ2)} (18) holds uniformly in 𝑓∈𝐶^∞₀(ℝ²).

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 𝜑^′′₁₂>0. Suppose that there exists a constant 𝐴 such that, for each 𝑥₁∈𝒟^∗, 𝜑^′′₁₂(𝑥₁,⋅)∈ℰ₂(𝐴); that is, 𝜑^′′₁₂𝑥₁,𝑦₁^1/2𝜑^′′₁₂𝑥₁,𝑦₂^1/2≤𝐴𝜑^′′₁₂𝑥₁,𝑦₁+𝑦₂2 (19) holds whenever 𝑦₁<𝑦₂ and [𝑦₁,𝑦₂]⊂𝒟_𝑥1. Let ℛ_𝜑 be the operator given by (14). Then, there exists a constant 𝐶 that depends only on 𝐴 such that ℛ_𝜑𝑓_{𝐿3,3/2(ℝ2)}≤𝐶𝑓_{𝐿3/2(ℝ2)} (20) holds uniformly in 𝑓∈𝐶^∞₀(ℝ²).

Proof of Theorem 8. To ease our notation, we let 𝜔∶=𝜑^′′₁₂. 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 𝐼𝐸₁,𝐸₂,𝐸₃∶=_ℝ2³_𝑗=1ℛ𝟙_𝐸𝑗𝑥₁,𝑥₂𝑑𝑥₁𝑑𝑥₂≤𝐶𝐸₁𝐸₂^1/2𝐸₃^1/2 (21) for all measurable subsets 𝐸₁, 𝐸₂, and 𝐸₃ of ℝ². We have 𝐼𝐸₁,𝐸₂,𝐸₃=_ℝ2³_{𝑗=1𝒟𝑥1}𝟙_𝐸𝑗𝑦_𝑗,𝑥₂+𝜑𝑥₁,𝑦_𝑗×𝜔𝑥₁,𝑦_𝑗^1/3𝑑𝑦_𝑗𝑑𝑥₁𝑑𝑥₂=_𝒟∗ℝ𝒜𝑥₂,𝑦₁;𝐸₂,𝐸₃𝟙_𝐸1𝑦₁,𝑥₂𝑑𝑥₂𝑑𝑦₁, (22) where 𝒜𝑥₂,𝑦₁;𝐸₂,𝐸₃∶=_𝒟𝑦1³_𝑗=2ℬ𝑥₁,𝑥₂,𝑦₁;𝐸_𝑗𝜔𝑥₁,𝑦₁^1/3𝑑𝑥₁,ℬ𝑥₁,𝑥₂,𝑦₁;𝐸∶=_𝒟𝑥1𝟙_𝐸𝑦,𝑥₂+𝜑𝑥₁,𝑦−𝜑𝑥₁,𝑦₁𝜔𝑥₁,𝑦^1/3𝑑𝑦. (23) By Schwarz inequality, it suffices to get an estimate _𝒟𝑦1ℬ𝑥₁,𝑥₂,𝑦₁;𝐸²𝜔𝑥₁,𝑦₁^1/3𝑑𝑥₁≤𝐶|𝐸| (24) uniformly in 𝑥₂, 𝑦₁, and 𝐸. By translation invariance of ℬ(𝑥₁,𝑥₂,𝑦₁;𝐸) in 𝑥₂ variable, it is enough to establish _𝒟𝑦1ℬ₁𝑥₁,𝑦₁;𝐸²𝜔𝑥₁,𝑦₁^1/3𝑑𝑥₁≤𝐶_𝒟𝑦1|𝐸|,ℬ₂𝑥₁,𝑦₁;𝐸²𝜔𝑥₁,𝑦₁^1/3𝑑𝑥₁≤𝐶|𝐸| (25)(26) uniformly in 𝑦₁ and 𝐸, whereℬ₁𝑥₁,𝑦₁;𝐸∶=_{𝒟𝑥1∩[𝑦1,∞)}𝟙_𝐸𝑦,𝜑𝑥₁,𝑦−𝜑𝑥₁,𝑦₁𝜔𝑥₁,𝑦^1/3ℬ𝑑𝑦,₂𝑥₁,𝑦₁;𝐸∶=_{𝒟𝑥1∩(−∞,𝑦1)}𝟙_𝐸𝑦,𝜑𝑥₁,𝑦−𝜑𝑥₁,𝑦₁𝜔𝑥₁,𝑦^1/3𝑑𝑦. (27)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 𝐽(𝑥₁,𝑦)∶=|𝜑^′₁(𝑥₁,𝑦)−𝜑^′₁(𝑥₁,𝑦₁)| for a given 𝑦₁∈𝒟_∗.

3.1. Estimate for  ℬ₁

We follow an approach by Oberlin [14]. Letting 𝐹∶=𝐹𝑥₁,𝑦₁;𝐸∶=𝑦∈𝒟_𝑥1∩𝑦₁,∞∶Γ𝑥₁,𝑦∈𝐸, (28) we have _{𝒟𝑥1∩[𝑦1,∞)}𝟙_𝐹𝑦𝜑^′₁𝑥₁,𝑦−𝜑^′₁𝑥₁,𝑦₁=𝑑𝑦_{𝒟𝑥1∩[𝑦1,∞)}𝟙_𝐹𝑦^𝑦_𝑦1𝜔𝑥₁=,𝑧𝑑𝑧𝑑𝑦_[𝑦1,∞)𝐹_𝑧𝜔𝑥₁,𝑧𝑑𝑧. (29) 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 _{𝒟𝑥1∩[𝑦1,∞)}𝟙_𝐹(𝑧)𝜔𝑥₁,𝑧^1/3𝑑𝑧≤2^2/3|𝐹|^1/3_{𝒟𝑥1∩[𝑦1,∞)}𝟙_𝐹𝑦𝐽𝑥₁,𝑦𝑑𝑦^1/3. (30) Combined with the monotonicity of 𝜔(𝑥₁,⋅), we obtain 𝜔𝑥₁,𝑦₁^1/3ℬ₁𝑥₁,𝑦₁;𝐸²=𝜔𝑥₁,𝑦₁^1/3_{𝒟𝑥1∩[𝑦1,∞)}𝟙_𝐹(𝑧)𝜔𝑥₁,𝑧^1/3𝑑𝑧²≤|𝐹|⁻¹_{𝒟𝑥1∩[𝑦1,∞)}𝟙_𝐹(𝑧)𝜔𝑥₁,𝑧^1/3𝑑𝑧³≤4_{𝒟𝑥1∩[𝑦1,∞)}𝟙_𝐹𝑦𝐽𝑥₁,𝑦𝑑𝑦. (31) An integration in 𝑥₁ provides (25).

3.2. Estimate forℬ₂

For fixed 𝑥₁ and 𝑦₁, we let 1𝜌=2𝐶(1)_{𝒟𝑥1∩(−∞,𝑦1)}𝟙_𝐹𝑦𝜔𝑥₁,𝑦^1/3𝑑𝑦, (32) where 𝐶(1) is the constant that appears in Lemma 2.2 in [11], which implies _{𝜔(𝑥1,𝑦1)1/3|𝑦−𝑦1|≥𝜌}𝟙_𝐹𝑦𝜔𝑥₁,𝑦^1/3≥1𝑑𝑦2_{𝒟𝑥1∩(−∞,𝑦1)}𝟙_𝐹𝑦𝜔𝑥₁,𝑦^1/3𝑑𝑦. (33) Since 𝜔(𝑥₁,⋅) is nondecreasing, we see LHSof(26)≤2_{𝒟∗𝒟𝑥1∩(−∞,𝑦1)}𝑦₁−𝑦𝜔𝑥₁,𝑦₁^2/3×𝜔𝑥₁,𝑦^1/3𝟙_𝐹𝑦𝑑𝑦𝑑𝑥₁≤2𝑐(𝐴)_{𝒟∗𝒟𝑥1∩(−∞,𝑦1)}𝐽𝑥₁,𝑦𝟙_𝐹𝑦𝑑𝑦𝑑𝑥₁=2𝑐(𝐴)_𝒟∗𝒟_{𝑥1∩(−∞,𝑦1)}𝟙_𝐸𝑥₁,𝑦𝑑𝑦𝑑𝑥₁=2𝑐(𝐴)𝐸₁. (34) 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 𝜑^′′₁₂>0. Suppose that there exists a constant 𝐴 such that, for each 𝑦∈𝒟_∗, 𝜑^′′₁₂(⋅,𝑦)∈ℰ₂(𝐴); that is, 𝜑^′′₁₂𝑥₁,𝑦^1/2𝜑^′′₁₂𝑥₂,𝑦^1/2≤𝐴𝜑^′′₁₂𝑥₁+𝑥₂2,𝑦₁ (35) 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 ℛ_𝜑𝑓_𝐿3(ℝ2)≤𝐶𝑓_{𝐿3/2,3(ℝ2)} (36) holds uniformly in 𝑓∈𝐶^∞₀(ℝ²).

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.