Abstract and Applied Analysis

Volume 2013, Article ID 582137, 5 pages

http://dx.doi.org/10.1155/2013/582137

## Uniform Estimates for Damped Radon Transform on the Plane

Department of Mathematics, Ajou University, Suwon 443-749, Republic of Korea

Received 13 May 2013; Accepted 22 August 2013

Academic Editor: Wenchang Sun

Copyright © 2013 Youngwoo Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 be a domain in and let For and , we write To avoid technical difficulties, we assume that, for and , and are finite intervals throughout the paper. For a function and a measurable function defined on , we consider the damped Radon transform 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 with a function 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 in , a locally integrable function , and a positive real number , we let An interesting subclass of is the collection , introduced in [13], of functions such that (1) is monotone, (2)whenever and ,

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 be an open interval in , and let be a function such that . Let be a nonnegative measurable function. Suppose that there exists a positive constant such that ; that is,
**
holds whenever and . Let be the operator given by
**
for . Then, there exists a constant that depends only on such that
**
holds uniformly in . *

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

Theorem 2 (Oberlin [11]). *Let be a function on an interval such that on and . Then, defined in (7) maps boundedly to with the operator norm depending only on . *

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 be a function on such that , and let be a nonnegative measurable function on . Suppose that there exists a positive constant such that, for each , ; that is,
**
holds whenever and . Let be the operator given by (3). Then, there exists a constant that depends only on 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 and suitable functions , , defined on ,with . As in the proof of Theorem 2.1 in [10], one can show that the estimate
holds uniformly in , , , , , and . 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 be a function such that . Suppose that there exists a constant such that, for each , ; that is,
**
holds whenever and . Let be the operator given by (14). Then, there exists a constant that depends only on such that
**
holds uniformly in . *

*Remark 6. *A duality argument shows the following.

Corollary 7. *Let be a function such that . Suppose that there exists a constant such that, for each , ; that is,
**
holds whenever and . Let be the operator given by (14). Then, there exists a constant that depends only on 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 be a function such that . Suppose that there exists a constant such that, for each , ; that is,
**
holds whenever and . Let be the operator given by (14). Then, there exists a constant that depends only on such that
**
holds uniformly in . *

*Proof of Theorem 8. *To ease our notation, we let . For a measurable subset of either or , we denote the Lebesgue measure and the characteristic function of by and , respectively.

By a well-known interpolation argument as in [2, 18], it suffices to establish the estimate
for all measurable subsets , , and of . We have
where
By Schwarz inequality, it suffices to get an estimate
uniformly in , , and . By translation invariance of in variable, it is enough to establish
uniformly in and , whereNotice that the map is one-to-one and has the absolute value of Jacobian determinant for a given .

##### 3.1. Estimate for

We follow an approach by Oberlin [14]. Letting we have Here, for , we denoted by the set . On the other hand, applying Hölder’s inequality as in [14], we get Combined with the monotonicity of , we obtain An integration in provides (25).

##### 3.2. Estimate for

For fixed and , we let where is the constant that appears in Lemma 2.2 in [11], which implies Since 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 be a function such that . Suppose that there exists a constant such that, for each , ; that is,
**
holds uniformly in . *

*Remark 11. *As is well known, if maps boundedly from to , then belongs to the convex hull of , and uniform estimates are possible only if . In the latter case, 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.

#### References

- J.-G. Bak, “An ${L}^{p}\text{-}{L}^{q}$ estimate for Radon transforms associated to polynomials,”
*Duke Mathematical Journal*, vol. 101, no. 2, pp. 259–269, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-G. Bak, D. M. Oberlin, and A. Seeger, “Two endpoint bounds for generalized Radon transforms in the plane,”
*Revista Matemática Iberoamericana*, vol. 18, no. 1, pp. 231–247, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. H. Lee, “Remarks on the degenerate Radon transform in ${R}^{2}$,”
*Proceedings of the American Mathematical Society*, vol. 129, no. 11, pp. 3373–3378, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Lee, “Endpoint ${L}^{p}-{L}^{q}$ estimates for degenerate Radon transforms in ${R}^{2}$ associated with real-analytic functions,”
*Mathematische Zeitschrift*, vol. 243, no. 2, pp. 217–241, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Lee, “Endpoint ${L}^{p}$-${L}^{q}$ estimates for some classes of degenerate Radon transforms in ${R}^{2}$,”
*Mathematical Research Letters*, vol. 11, no. 1, pp. 85–101, 2004. View at Google Scholar · View at MathSciNet - D. H. Phong and E. M. Stein, “Models of degenerate Fourier integral operators and Radon transforms,”
*Annals of Mathematics*, vol. 140, no. 3, pp. 703–722, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Seeger, “Degenerate Fourier integral operators in the plane,”
*Duke Mathematical Journal*, vol. 71, no. 3, pp. 685–745, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Seeger, “Radon transforms and finite type conditions,”
*Journal of the American Mathematical Society*, vol. 11, no. 4, pp. 869–897, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. W. Yang, “${L}^{p}$ improving estimates for some classes of Radon transforms,”
*Transactions of the American Mathematical Society*, vol. 357, no. 10, pp. 3887–3903, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Choi, “Convolution estimates related to space curves,”
*Journal of Inequalities and Applications*, vol. 2011, no. 91, pp. 1–6, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - D. M. Oberlin, “Convolution with measures on flat curves in low dimensions,”
*Journal of Functional Analysis*, vol. 259, no. 7, pp. 1799–1815, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. W. Drury, “Degenerate curves and harmonic analysis,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 108, no. 1, pp. 89–96, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-G. Bak, D. M. Oberlin, and A. Seeger, “Restriction of Fourier transforms to curves. II. Some classes with vanishing torsion,”
*Journal of the Australian Mathematical Society*, vol. 85, no. 1, pp. 1–28, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. M. Oberlin, “Convolution with affine arclength measures in the plane,”
*Proceedings of the American Mathematical Society*, vol. 127, no. 12, pp. 3591–3592, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. W. Drury and K. H. Guo, “Convolution estimates related to surfaces of half the ambient dimension,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 110, no. 1, pp. 151–159, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. M. Oberlin, “Two estimates for curves in the plane,”
*Proceedings of the American Mathematical Society*, vol. 132, no. 11, pp. 3195–3201, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Christ, “On the restriction of the Fourier transform to curves: endpoint results and the degenerate case,”
*Transactions of the American Mathematical Society*, vol. 287, no. 1, pp. 223–238, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. M. Oberlin, “Convolution with measures on hypersurfaces,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 129, no. 3, pp. 517–526, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Stovall, “Endpoint bounds for a generalized Radon transform,”
*Journal of the London Mathematical Society*, vol. 80, no. 2, pp. 357–374, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet