# Uniform Estimates for Damped Radon Transform on the Plane

Authors: Youngwoo Choi
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction

Let𝒟 be a domain in ℝ2 and let
(1)𝒟*:={x1∈ℝ:(x1,y)∈𝒟forsomey∈ℝ},𝒟*:={y∈ℝ:(x1,y)∈𝒟forsomex1∈ℝ}.
For x1∈𝒟* and y∈𝒟*, we write
(2)𝒟x1:={z∈ℝ:(x1,z)∈𝒟},𝒟y:={z∈ℝ:(z,y)∈𝒟}.
To avoid technical difficulties, we assume that, for x1∈𝒟* and y∈𝒟*, 𝒟x1 and 𝒟y 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 ℛφ,ω defined by
(3)ℛφ,ωf(x1,x2)∶=∫𝒟x1‍f(y,x2+φ(x1,y))ω(x1,y)dy
for f∈C0∞(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 φ12′′(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 intervalJ1 in ℝ, a locally integrable function Φ:J1→ℝ+, and a positive real number A, we let
(4)𝔊(Φ,A):={ω:J1⟶ℝ+∣ω(s1)ω(s2)≤As2-s1∫s1s2‍Φ(s)dswhenevers1<s2and[s1,s2]⊂J1(ω:J1→ℝ+ω(s1)ω(s2))},ℰ1(A):={Φ:J⟶ℝ+∣Φ∈𝔊(Φ,A)}.
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)
whenevers1<s2 and [s1,s2]⊂J,
(5)Φ(s1)Φ(s2)≤AΦ(s1+s22).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.
LetJ 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,
(6)ω(s1)1/2ω(s2)1/2≤As2-s1∫s1s2‍ϕ′′(v)dv
holds whenever s1<s2 and [s1,s2]⊂J. Let 𝒮ϕ be the operator given by
(7)𝒮ϕf(x1,x2)=∫J‍f(x1-s,x2-ϕ(s))ω1/3(s)ds
for f∈C0∞(ℝ2). Then, there exists a constant C that depends only on A such that
(8)∥𝒮ϕf∥L3(ℝ2)≤C∥f∥L3/2(ℝ2)
holds uniformly in f∈C0∞(ℝ2).Regarding the endpoint Lorentz space estimates, the following result due to Oberlin is available.Theorem 2 (Oberlin [11]).
Letϕ be a C2 function on an interval J such that ϕ′′>0 on J and ϕ′′∈ℰ2(A). Then, 𝒮ϕ defined in (7) maps L3/2,3(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 Section2, 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.
Letφ be a C2 function on 𝒟 such that φ12′′>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(φ12′′(x1,·),A); that is,
(9)ω(x1,y1)1/2ω(x1,y2)1/2≤Ay2-y1∫y1y2‍φ12′′(x1,z)dz
holds whenever y1<y2 and [y1,y2]⊂𝒟x1. Let ℛφ,ω be the operator given by (3). Then, there exists a constant C that depends only on A such that
(10)∥ℛφ,ωf∥L3(ℝ2)≤C∥f∥L3/2(ℝ2)
holds uniformly in f∈C0∞(ℝ2).Proof of Theorem3.
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
(11)∥ℛφ,ωf∥33=∫ℝ‍∫𝒟*‍∏j=13‍(∫𝒟x1‍f(y1,x2+φ(x1,yj))×ω1/3(x1,yj)dyj∫𝒟x1‍f(y1,x2+φ(x1,yj)))dx1dx2=∭𝒟*[𝒢(f(y1,·),f(y2,·),f(y3,·))]×(y1,y2,y3)dy1dy2dy3,
where for y1,y2,y3∈𝒟* and suitable functions g1, g2, g3 defined on ℝ,(12)[𝒢(g1,g2,g3)](y1,y2,y3):=∫ℝ‍∫𝒟y1,y2,y3‍∏j=13‍[gj(x2+φ(x1,yj))ω1/3(x1,yj)]dx1dx2with 𝒟y1,y2,y3:=⋂j=13‍𝒟yj. As in the proof of Theorem 2.1 in [10], one can show that the estimate
(13)|[𝒢(g1,g2,g3)](y1,y2,y3)|≤C∥g1∥L3/2(ℝ)∥g2∥L3/2(ℝ)∥g3∥L3/2(ℝ)|(y1-y2)(y1-y3)(y2-y3)|1/3
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 whichω=φ12′′ provides a uniform estimate for the damped plane Radon transform. We write
(14)ℛφf(x1,x2)=∫𝒟x1‍f(y,x2+φ(x1,y))φ12′′(x1,x2,y)1/3dy
for f∈C0∞(ℝ2).Corollary 5.
Letφ:𝒟→ℝ be a C2 function such that φ12′′>0. Suppose that there exists a constant A such that, for each x1∈𝒟*, φ12′′(x1,·)∈ℰ1(A); that is,
(15)φ12′′(x1,y1)1/2φ12′′(x1,y2)1/2≤Ay2-y1∫y1y2‍φ12′′(x1,z)dz
holds whenever y1<y2 and [y1,y2]⊂𝒟x1. Let ℛφ be the operator given by (14). Then, there exists a constant C that depends only on A such that
(16)∥ℛφf∥L3(ℝ2)≤C∥f∥L3/2(ℝ2)
holds uniformly in f∈C0∞(ℝ2).Remark 6.
A duality argument shows the following.Corollary 7.
Letφ:𝒟→ℝ be a C2 function such that φ12′′>0. Suppose that there exists a constant A such that, for each y1∈𝒟*, φ12′′(·,y1)∈ℰ1(A); that is,
(17)φ12′′(x1,y1)1/2φ12′′(x2,y1)1/2≤Ax2-x1∫x1x2‍φ12′′(z,y1)dz
holds whenever x1<x2 and [x1,x2]⊂𝒟y1. Let ℛφ be the operator given by (14). Then, there exists a constant C that depends only on A such that
(18)∥ℛφf∥L3(ℝ2)≤C∥f∥L3/2(ℝ2)
holds uniformly in f∈C0∞(ℝ2).


## 3. Endpoint Lorentz Estimates

Under somewhat stronger condition, estimates in Section2 can be improved. Namely, we have the following.Theorem 8.
Letφ:𝒟→ℝ be a C2 function such that φ12′′>0. Suppose that there exists a constant A such that, for each x1∈𝒟*, φ12′′(x1,·)∈ℰ2(A); that is,
(19)φ12′′(x1,y1)1/2φ12′′(x1,y2)1/2≤Aφ12′′(x1,(y1+y2)2)
holds whenever y1<y2 and [y1,y2]⊂𝒟x1. Let ℛφ be the operator given by (14). Then, there exists a constant C that depends only on A such that
(20)∥ℛφf∥L3,3/2(ℝ2)≤C∥f∥L3/2(ℝ2)
holds uniformly in f∈C0∞(ℝ2).Proof of Theorem8.
To ease our notation, we letω:=φ12′′. For a measurable subset E of either ℝ or ℝ2, we denote the Lebesgue measure and the characteristic function of E by |E| and 1E, respectively.
By a well-known interpolation argument as in [2, 18], it suffices to establish the estimate
(21)I(E1,E2,E3)≔∬ℝ2‍∏j=13‍[ℛ1Ej(x1,x2)]dx1dx2≤C|E1||E2|1/2|E3|1/2
for all measurable subsets E1, E2, and E3 of ℝ2. We have
(22)I(E1,E2,E3)=∬ℝ2‍∏j=13‍[∫𝒟x1‍1Ej(yj,x2+φ(x1,yj))×ω(x1,yj)1/3dyj∫𝒟x1‍1Ej(yj,x2+φ(x1,yj))]dx1dx2=∫𝒟*‍∫ℝ‍[𝒜(x2,y1;E2,E3)]1E1(y1,x2)dx2dy1,
where
(23)𝒜(x2,y1;E2,E3):=∫𝒟y1‍∏j=23‍[ℬ~(x1,x2,y1;Ej)]ω(x1,y1)1/3dx1,ℬ~(x1,x2,y1;E):=∫𝒟x1‍1E(y,x2+φ(x1,y)-φ(x1,y1))ω(x1,y)1/3dy.
By Schwarz inequality, it suffices to get an estimate
(24)∫𝒟y1‍[ℬ~(x1,x2,y1;E)]2ω(x1,y1)1/3dx1≤C|E|
uniformly in x2, y1, and E. By translation invariance of ℬ~(x1,x2,y1;E) in x2 variable, it is enough to establish
(25)∫𝒟y1‍[ℬ1(x1,y1;E)]2ω(x1,y1)1/3dx1≤C|E|,(26)∫𝒟y1‍[ℬ2(x1,y1;E)]2ω(x1,y1)1/3dx1≤C|E|
uniformly in y1 and E, where(27)ℬ1(x1,y1;E):=∫𝒟x1∩[y1,∞)‍1E(y,φ(x1,y)-φ(x1,y1))ω(x1,y)1/3dy,ℬ2(x1,y1;E):=∫𝒟x1∩(-∞,y1)‍1E(y,φ(x1,y)-φ(x1,y1))ω(x1,y)1/3dy.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):=|φ1′(x1,y)-φ1′(x1,y1)| for a given y1∈𝒟*.


### 3.1. Estimate forℬ1

We follow an approach by Oberlin [14]. Letting
(28)F:=F(x1,y1;E):={y∈𝒟x1∩[y1,∞):Γ(x1,y)∈E},
we have
(29)∫𝒟x1∩[y1,∞)‍1F(y)(φ1′(x1,y)-φ1′(x1,y1))dy=∫𝒟x1∩[y1,∞)‍1F(y)∫y1y‍ω(x1,z)dzdy=∫[y1,∞)‍|Fz|ω(x1,z)dz.
Here, for z>y1, we denoted by Fz the set F∩[z,∞). On the other hand, applying Hölder’s inequality as in [14], we get
(30)∫𝒟x1∩[y1,∞)‍1F(z)ω(x1,z)1/3dz≤22/3|F|1/3(∫𝒟x1∩[y1,∞)‍1F(y)J(x1,y)dy)1/3.
Combined with the monotonicity of ω(x1,·), we obtain
(31)ω(x1,y1)1/3[ℬ1(x1,y1;E)]2=ω(x1,y1)1/3(∫𝒟x1∩[y1,∞)‍1F(z)ω(x1,z)1/3dz)2≤|F|-1(∫𝒟x1∩[y1,∞)‍1F(z)ω(x1,z)1/3dz)3≤4∫𝒟x1∩[y1,∞)‍1F(y)J(x1,y)dy.
An integration in x1 provides (25).


### 3.2. Estimate forℬ2

For fixedx1 and y1, we let
(32)ρ=12C(1)∫𝒟x1∩(-∞,y1)‍1F(y)ω(x1,y)1/3dy,
where C(1) is the constant that appears in Lemma 2.2 in [11], which implies
(33)∫ω(x1,y1)1/3|y-y1|≥ρ‍1F(y)ω(x1,y)1/3dy≥12∫𝒟x1∩(-∞,y1)‍1F(y)ω(x1,y)1/3dy.
Since ω(x1,·) is nondecreasing, we see
(34)LHSof(26)≤2∫𝒟*‍∫𝒟x1∩(-∞,y1)‍|y1-y|ω(x1,y1)2/3×ω(x1,y)1/31F(y)dydx1≤2c(A)∫𝒟*‍∫𝒟x1∩(-∞,y1)‍J(x1,y)1F(y)dydx1=2c(A)∫𝒟*‍∫𝒟~x1∩(-∞,y1)‍1E(x1,y)dydx1=2c(A)|E1|.
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.
Letφ:𝒟→ℝ be a C2 function such that φ12′′>0. Suppose that there exists a constant A such that, for each y∈𝒟*, φ12′′(·,y)∈ℰ2(A); that is,
(35)φ12′′(x1,y)1/2φ12′′(x2,y)1/2≤Aφ12′′((x1+x2)2,y1)
holds whenever x1<x2 and [x1,x2]⊂𝒟y. Let ℛφ be the operator given by (14). Then, there exists a constant C that depends only on A such that
(36)∥ℛφf∥L3(ℝ2)≤C∥f∥L3/2,3(ℝ2)
holds uniformly in f∈C0∞(ℝ2).Remark 11.
As is well known, ifℛφ maps boundedly from Lp,u(ℝ2) to Lq,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].


## 3.1. Estimate forℬ1

We follow an approach by Oberlin [14]. Letting
(28)F:=F(x1,y1;E):={y∈𝒟x1∩[y1,∞):Γ(x1,y)∈E},
we have
(29)∫𝒟x1∩[y1,∞)‍1F(y)(φ1′(x1,y)-φ1′(x1,y1))dy=∫𝒟x1∩[y1,∞)‍1F(y)∫y1y‍ω(x1,z)dzdy=∫[y1,∞)‍|Fz|ω(x1,z)dz.
Here, for z>y1, we denoted by Fz the set F∩[z,∞). On the other hand, applying Hölder’s inequality as in [14], we get
(30)∫𝒟x1∩[y1,∞)‍1F(z)ω(x1,z)1/3dz≤22/3|F|1/3(∫𝒟x1∩[y1,∞)‍1F(y)J(x1,y)dy)1/3.
Combined with the monotonicity of ω(x1,·), we obtain
(31)ω(x1,y1)1/3[ℬ1(x1,y1;E)]2=ω(x1,y1)1/3(∫𝒟x1∩[y1,∞)‍1F(z)ω(x1,z)1/3dz)2≤|F|-1(∫𝒟x1∩[y1,∞)‍1F(z)ω(x1,z)1/3dz)3≤4∫𝒟x1∩[y1,∞)‍1F(y)J(x1,y)dy.
An integration in x1 provides (25).


## 3.2. Estimate forℬ2

For fixedx1 and y1, we let
(32)ρ=12C(1)∫𝒟x1∩(-∞,y1)‍1F(y)ω(x1,y)1/3dy,
where C(1) is the constant that appears in Lemma 2.2 in [11], which implies
(33)∫ω(x1,y1)1/3|y-y1|≥ρ‍1F(y)ω(x1,y)1/3dy≥12∫𝒟x1∩(-∞,y1)‍1F(y)ω(x1,y)1/3dy.
Since ω(x1,·) is nondecreasing, we see
(34)LHSof(26)≤2∫𝒟*‍∫𝒟x1∩(-∞,y1)‍|y1-y|ω(x1,y1)2/3×ω(x1,y)1/31F(y)dydx1≤2c(A)∫𝒟*‍∫𝒟x1∩(-∞,y1)‍J(x1,y)1F(y)dydx1=2c(A)∫𝒟*‍∫𝒟~x1∩(-∞,y1)‍1E(x1,y)dydx1=2c(A)|E1|.
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.
Letφ:𝒟→ℝ be a C2 function such that φ12′′>0. Suppose that there exists a constant A such that, for each y∈𝒟*, φ12′′(·,y)∈ℰ2(A); that is,
(35)φ12′′(x1,y)1/2φ12′′(x2,y)1/2≤Aφ12′′((x1+x2)2,y1)
holds whenever x1<x2 and [x1,x2]⊂𝒟y. Let ℛφ be the operator given by (14). Then, there exists a constant C that depends only on A such that
(36)∥ℛφf∥L3(ℝ2)≤C∥f∥L3/2,3(ℝ2)
holds uniformly in f∈C0∞(ℝ2).Remark 11.
As is well known, ifℛφ maps boundedly from Lp,u(ℝ2) to Lq,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].



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