Uniform Lorentz norm estimates for convolution operators
 Youngwoo Choi^{1}Email author
https://doi.org/10.1186/1029242X2014256
© Choi; licensee Springer 2014
Received: 8 January 2014
Accepted: 1 July 2014
Published: 22 July 2014
Abstract
Uniform endpoint Lorentz norm improving estimates for convolution operators with affine arclength measure supported on simple plane curves are established. The estimates hold for a wide class of simple curves, and the condition is stated in terms of averages of the square of the affine arclength weight, extending previously known results.
MSC:44A35, 42B35.
Keywords
1 Introduction
for $f\in {C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{2})$. Here and in what follows, we denote $\omega (t):={({\varphi}^{\u2033}(t))}^{1/3}$. Curves of the form $(t,\varphi (t))$ are said to be simple according to Drury and Marshall [1]. The measure $\omega (t)\phantom{\rule{0.2em}{0ex}}dt$ supported on the curve $(t,\varphi (t))$ is known as the affine arclength measure, which is based on the affine arclength parameter as in [2], and was introduced by Drury and Marshall [1] in dealing with the Fourier restriction problem related to curves, and later by Drury [3] in studying convolution operators with measures supported on curves. We refer interested readers to [2–4] for the relevance of affine geometry in this subject. One big benefit of using the affine arclength measure in place of the Euclidean arclength measure $\sqrt{1+{\varphi}^{\prime}{(t)}^{2}}\phantom{\rule{0.2em}{0ex}}dt$ has been its effect of mitigating degeneracies and it is believed that various uniform sharp estimates hold for a wide class of curves.
As is well known, the typeset $\mathcal{S}=\{({p}^{1},{q}^{1}):\mathcal{T}\text{is bounded from}{L}^{p}({\mathbb{R}}^{2})\text{to}{L}^{q}({\mathbb{R}}^{2})\}$ of is contained in the convex hull of $\{(0,0),(1,1),(2/3,1/3)\}$ and uniform estimates in a, b, and ϕ are expected only for $(1/p,1/q)=(2/3,1/3)$. Many conditions to guarantee optimal uniform ${L}^{3/2}$${L}^{3}$ estimates have been known so far. See [3, 5–12] for example. Among other things, the author proved the following.
Theorem 1.1 (Choi [12])
holds uniformly in $f\in {C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{2})$.
Under somewhat stronger assumptions on $\varphi (t)$ or $\omega (t)$, the endpoint Lebesgue norm estimate aforementioned can be improved to optimal Lorentz norm estimates, namely from ${L}^{3/2}({\mathbb{R}}^{2})$ into ${L}^{3,3/2}({\mathbb{R}}^{2})$ and ${L}^{3/2,3}({\mathbb{R}}^{2})$ into ${L}^{3}({\mathbb{R}}^{2})$. We refer interested readers to [6, 8, 10, 11] for known sufficient conditions for optimal and nearly optimal Lorentz norm estimates. Most importantly, Oberlin established the following uniform optimal Lorentz norm improving estimates.
Theorem 1.2 (Oberlin [11])
for all $f\in {C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{2})$, where C is a constant depending only on A.
For the proof of the optimality, see [13] by Stovall along with [8] by Bak et al. It is interesting to ask if the condition in Theorem 1.2 can be relaxed to cover more general curves. Based on an ingenious argument of Oberlin in [11], the author aims to establish a uniform optimal Lorentz norm improving estimate under a condition on averages of the square of $\omega (t)$. The average condition is a slightly stronger version of that in Theorem 1.1, and yet covers most simple plane curves studied up to now including those in Theorem 1.2.
This paper is organized as follows: in the following section, conditions on $\omega (t)$ are introduced and the main theorem is stated. The last section is devoted to the proof of the main theorem. As usual, absolute constants may grow from line to line.
2 Statement of the main theorem
Before we state our main result, we introduce certain conditions on functions defined on intervals.
An interesting subclass of ${\mathcal{E}}_{p}({2}^{1/p}A)$, $0<p<\mathrm{\infty}$, was introduced by Bak et al. [14] in studying Fourier restriction estimates related to degenerate curves.
Definition 2.2 For an interval J and a positive real number A, a function $\mathrm{\Phi}:J\to {\mathbb{R}}^{+}$ is said to be a member of $\tilde{\mathcal{E}}(A)$ if

Φ is monotone; and

whenever ${t}_{1}<{t}_{2}$ and $[{t}_{1},{t}_{2}]\subset J$,$\sqrt{\mathrm{\Phi}({t}_{1})\mathrm{\Phi}({t}_{2})}\le A\mathrm{\Phi}(({t}_{1}+{t}_{2})/2)$
holds.
The condition (1.2) can be rewritten as $\omega \in \tilde{\mathcal{E}}(A)$.
 1.
It is a simple matter to check:

$\tilde{\mathcal{E}}(A)\subset {\mathcal{E}}_{p}({2}^{1/p}A)$ for all $p\in (0,\mathrm{\infty})$;

$\mathrm{\Phi}\in {\mathcal{E}}_{p}(A)$ if and only if ${\mathrm{\Phi}}^{p}\in {\mathcal{E}}_{1}({A}^{p})$;

$\mathrm{\Phi}\in {\mathcal{E}}_{p}(A)$ implies $\lambda \mathrm{\Phi}\in {\mathcal{E}}_{p}(A)$ for all $\lambda >0$; and

$\mathrm{\Phi}\in {\mathcal{E}}_{p}(A)$ implies $\mathrm{\Phi}(a\cdot +b)\in {\mathcal{E}}_{p}(A)$ for all $(a,b)\in \mathbb{R}\setminus \{0\}\times \mathbb{R}$.
 2.
If $0<{p}_{1}<{p}_{2}<\mathrm{\infty}$, $\mathrm{\Phi}:J\to {\mathbb{R}}^{+}\in {\mathcal{E}}_{{p}_{1}}(A)$, and $\mathrm{\Phi}\in {L}_{\mathrm{loc}}^{{p}_{2}}(J)$, then $\mathrm{\Phi}\in {\mathcal{E}}_{{p}_{2}}(A)$ by Hölder’s inequality.
 3.
The class $\tilde{\mathcal{E}}(1)$ is essentially the class of logarithmically concave functions, which already encompasses many useful examples. Simplest examples are the exponential function and $\mathrm{\Phi}(t)={t}^{\alpha}$, $t>0$, for $\alpha \ge 0$. More interesting example is the function $\mathrm{\Phi}(t)={e}^{1/t}$, $t>0$, which models a curve ‘flat’ at the origin. A hierarchy of flatter functions that belong to $\tilde{\mathcal{E}}(1)$ was constructed by Bak et al. [14].
 4.
For a polynomial $p(t)$ of degree N, $p(t)$ belongs to $\tilde{\mathcal{E}}({2}^{N/2})$ after (possibly) decomposing the real line into at most ${3}^{N/2}$ intervals.
 5.
Nevertheless, there are functions that belong to ${\mathcal{E}}_{p}(A)$ but do not belong to $\tilde{\mathcal{E}}({A}^{\prime})$ for any ${A}^{\prime}>0$. Two examples of curves that our result covers that are not covered in [11] can be constructed with the aid of the examples given below.
which implies ${\mathrm{\Phi}}_{\beta}\in {\mathcal{E}}_{1}(2)$. In view of Remark 2.3, given $\beta >0$, ${\mathrm{\Phi}}_{\beta}\in {\mathcal{E}}_{p}({2}^{1/p})$ if $p\ge 2/\beta $. One can easily see ${\mathrm{\Phi}}_{\beta}\notin \tilde{\mathcal{E}}({A}^{\prime})$ for any ${A}^{\prime}>0$ and $\beta >0$.
Example 2.5 Consider $\mathrm{\Phi}:(0,\mathrm{\infty})\to {\mathbb{R}}^{+}$ given by $\mathrm{\Phi}(t)={(2t)}^{1/2}{e}^{{t}^{2}}$. Then we have $\sqrt{\mathrm{\Phi}(t)\mathrm{\Phi}(1)}\sim {t}^{1/4}{e}^{{t}^{2}/2}$ and $\mathrm{\Phi}((t+1)/2)=O({t}^{1/2}{e}^{{t}^{2}/3})$ as $t\to \mathrm{\infty}$, which clearly implies $\mathrm{\Phi}\notin \tilde{\mathcal{E}}(A)$ for all $A>0$. On the other hand, $\mathrm{\Phi}\in {\mathcal{E}}_{2}(1)$ by the following.
Proposition 2.6 Let $\psi :J\to \mathbb{R}$. Suppose that ${\psi}^{\prime}\in {\mathcal{E}}_{1}(A)$ for some $A>0$. Then the function Φ given by $\mathrm{\Phi}(t)={({\psi}^{\prime})}^{1/p}(t)exp(\psi (t))$ belongs to ${\mathcal{E}}_{p}({A}^{1/p})$ for $0<p<\mathrm{\infty}$.
By taking the p th root we obtain the desired estimate. □
We are now ready to state the main theorem of this paper.
holds uniformly in $f\in {C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{2})$.
Remark 2.8 Some remarks are in order.

In view of Remark 2.3, Proposition 2.6, Example 2.4 and Example 2.5, the condition $\omega \in \tilde{\mathcal{E}}(A)$ is strictly stronger than the condition $\omega \in {\mathcal{E}}_{2}(\sqrt{2}A)$ in Theorem 2.7, and therefore our result improves Theorem 1.2.

An explicit example is also available. Consider $\varphi (t)={t}^{1/2}exp({t}^{2})$ defined for $t\in (c,\mathrm{\infty})$, where c is a large constant. A simple calculation shows $\omega (t)\sim {t}^{1/2}exp({t}^{2}/3)$. By Proposition 2.6, $\omega \in {\mathcal{E}}_{2}(A)$ for some $A>0$. Thus, the corresponding operator satisfies endpoint Lorentz estimates (2.1) and (2.2) by Theorem 2.7, but Theorem 1.2 is not directly applicable.

It is not known whether $\omega \in {\mathcal{E}}_{2}(A)$ in Theorem 2.7 can be further relaxed to $\omega \in {\mathcal{E}}_{p}(A)$ for some $p>2$. More generally, one can ask for the optimal p such that $\omega \in {\mathcal{E}}_{p}(A)$ guarantees the boundedness of from ${L}^{\frac{3}{2},q}({\mathbb{R}}^{2})$ to ${L}^{3,r}({\mathbb{R}}^{2})$ for given $q\le r$.
3 Proof of the main theorem
Before we prove the theorem, we begin with a couple of lemmas.
whenever ${t}_{1}<{t}_{2}$ and ${t}^{\ast}\in [{t}_{1},{t}_{2}]\subset J$.
By taking the $2/3$th power, we obtain the desired estimate. □
The following lemma, which is nearly a triviality, generalizes a version of Lemma 2.2 in [11].
which finishes the proof. □
Proof of Theorem 2.7 It suffices to prove (2.1) by duality. We may further assume, without loss of generality, $\mathrm{\infty}<a<b<\mathrm{\infty}$, since a uniform estimate independent of a and b will allow us a suitable limiting argument. For a measurable subset E of either ℝ or ${\mathbb{R}}^{2}$, we denote the Lebesgue measure and the characteristic function of E by $E$ and , respectively. We also write $\gamma (t)=(t,\varphi (t))$.
Letting $\mathrm{\Omega}=\{({t}_{1},{t}_{2})\in \mathrm{\Delta}:\gamma ({t}_{1})\gamma ({t}_{2})\in E\}$ and making a change of variables, we obtain the desired estimate (3.2). □
Declarations
Acknowledgements
The author is grateful to the anonymous referee for valuable comments. This paper was completed with Ajou University research fellowship of 2013.
Authors’ Affiliations
References
 Drury SW, Marshall BP: Fourier restriction theorems for curves with affine and Euclidean arclengths. Math. Proc. Camb. Philos. Soc. 1985,97(1):111–125. 10.1017/S0305004100062654MathSciNetView ArticleMATHGoogle Scholar
 Guggenheimer HW: Differential Geometry. McGrawHill, New York; 1963.MATHGoogle Scholar
 Drury SW: Degenerate curves and harmonic analysis. Math. Proc. Camb. Philos. Soc. 1990,108(1):89–96. 10.1017/S0305004100068973MathSciNetView ArticleMATHGoogle Scholar
 Oberlin DM: Affine dimension: measuring the vestiges of curvature. Mich. Math. J. 2003,51(1):13–26. 10.1307/mmj/1049832890MathSciNetView ArticleMATHGoogle Scholar
 Choi Y: Convolution operators with the affine arclength measure on plane curves. J. Korean Math. Soc. 1999,36(1):193–207.MathSciNetMATHGoogle Scholar
 Oberlin DM: Convolution with measures on hypersurfaces. Math. Proc. Camb. Philos. Soc. 2000,129(3):517–526. 10.1017/S0305004100004552MathSciNetView ArticleMATHGoogle Scholar
 Oberlin DM: Two estimates for curves in the plane. Proc. Am. Math. Soc. 2004,132(11):3195–3201. 10.1090/S0002993904076105MathSciNetView ArticleMATHGoogle Scholar
 Bak JG, Oberlin DM, Seeger A: Two end point bounds for generalized Radon transforms in the plane. Rev. Mat. Iberoam. 2002,18(1):231–247.MathSciNetView ArticleMATHGoogle Scholar
 Oberlin DM: Convolution with measures on polynomial curves. Math. Scand. 2002,90(1):126–138.MathSciNetMATHGoogle Scholar
 Dendrinos S, Laghi N, Wright J:Universal ${L}^{p}$ improving for averages along polynomial curves in low dimensions. J. Funct. Anal. 2009,257(5):1355–1378. 10.1016/j.jfa.2009.05.011MathSciNetView ArticleMATHGoogle Scholar
 Oberlin DM: Convolution with measures on flat curves in low dimensions. J. Funct. Anal. 2010,259(7):1799–1815. 10.1016/j.jfa.2010.05.008MathSciNetView ArticleMATHGoogle Scholar
 Choi Y: Convolution estimates related to space curves. J. Inequal. Appl. 2011., 2011: Article ID 91Google Scholar
 Stovall B: Endpoint bounds for a generalized Radon transform. J. Lond. Math. Soc. 2009,80(2):357–374. 10.1112/jlms/jdp033MathSciNetView ArticleMATHGoogle Scholar
 Bak JG, Oberlin DM, Seeger A: Restriction of Fourier transforms to curves. II. Some classes with vanishing torsion. J. Aust. Math. Soc. 2008,85(1):1–28. 10.1017/S1446788708000578MathSciNetView ArticleMATHGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.