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Uniform Lorentz norm estimates for convolution operators
Journal of Inequalities and Applications volume 2014, Article number: 256 (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.
1 Introduction
Let \varphi :(a,b)\to \mathbb{R} be a {C}^{2} function such that {\varphi}^{\u2033}(t)\ge 0 for all t\in (a,b). In this paper, we consider the convolution operator given by
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])
Let J be an open interval in ℝ, and \varphi :J\to \mathbb{R} be a {C}^{2} function such that {\varphi}^{\u2033}\ge 0. Suppose that there exists a positive constant A such that
holds whenever {t}_{1}<{t}_{2} and [{t}_{1},{t}_{2}]\subset J. Let be the operator defined as in (1.1). Then there exists a constant C that depends only on A such that
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])
Let J be an open interval. Suppose that \omega (t) is monotone increasing and that there exists a positive constant A such that
holds whenever {t}_{1}<{t}_{2} and [{t}_{1},{t}_{2}]\subset J. Then the operator given by (1.1) satisfies
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.
Definition 2.1 Let 0<p<\mathrm{\infty}. For an interval {J}_{1} in ℝ, a locally {L}^{p} function \mathrm{\Phi}:{J}_{1}\to {\mathbb{R}}^{+}, and a positive real number A, we let
and we let
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).
Remark 2.3 It seems appropriate to mention some properties of {\mathcal{E}}_{p}(A) and \tilde{\mathcal{E}}(A) mentioned above.

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.
Example 2.4 Consider {\mathrm{\Phi}}_{\beta}(t)={t}^{\beta}, t>0, for \beta \ge 2. Then, for given 0<{t}_{1}<{t}_{2}<\mathrm{\infty}, we have by a change of variable
where \lambda :={t}_{2}/{t}_{1}>1. Since {t}^{\beta 2} is logarithmically concave, we see
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}.
Proof Let {t}_{1}<{t}_{2}. Since {\psi}^{\prime}\in {\mathcal{E}}_{1}(A), we have
by the fundamental theorem of calculus and the assumption on {\psi}^{\prime}(t). A change of variable gives
From
for all a<b, we see
Altogether, we obtain
By taking the p th root we obtain the desired estimate. □
We are now ready to state the main theorem of this paper.
Theorem 2.7 Let \mathrm{\infty}\le a<b\le \mathrm{\infty}, and let \varphi :(a,b)\to \mathbb{R} be a {C}^{2} function such that {\varphi}^{\u2033}\ge 0 on the interval. Suppose that there exists a positive constant A such that \omega \in {\mathcal{E}}_{2}(A), i.e.
holds whenever a<{t}_{1}<{t}_{2}<b. Let be the operator given by (1.1). Then there exists a constant C that depends only on A such that
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.
Lemma 3.1 Let J be an interval in ℝ, and let \omega :J\to {\mathbb{R}}_{+} be a continuous function such that \omega \in {\mathcal{E}}_{2}(A) for some A>0, i.e.
holds whenever {t}_{1}<{t}_{2} and [{t}_{1},{t}_{2}]\subset J. Then the following holds:
whenever {t}_{1}<{t}_{2} and {t}^{\ast}\in [{t}_{1},{t}_{2}]\subset J.
Proof of Lemma 3.1 Let {t}^{\ast}\in [{t}_{1},{t}_{2}]\subset J. From
we obtain
by hypothesis and Hölder’s inequality. Applying Hardy’s inequality twice gives us
and so we obtain
By taking the 2/3th power, we obtain the desired estimate. □
The following lemma, which is nearly a triviality, generalizes a version of Lemma 2.2 in [11].
Lemma 3.2 Suppose F is nonnegative and continuous on some interval [a,b]. For t\in [a,b], we let \tilde{F}(t):={max}_{[t,b]}F, and for \rho >0, we let
Then we have
Proof of Lemma 3.2 Observe that the function t\mapsto \tilde{F}(t)(bt) is a monotone decreasing function. Let \rho >0 be given. Since b\in {E}_{\rho}, {E}_{\rho} is nonempty. Let {t}_{\ast}:=inf{E}_{\rho}. Then we have \tilde{F}({t}_{\ast})(b{t}_{\ast})\le \rho. From this, we obtain
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)).
By a wellknown interpolation argument [7, 8], it suffices to show that
holds for measurable sets E\subset {\mathbb{R}}^{2}. In view of the simple identities
where \overline{\gamma}(t):=(t,\overline{\varphi}(t)), \overline{\varphi}(t):=\varphi (a+bt), \overline{\omega}(t):={({\overline{\varphi}}^{\u2033}(t))}^{1/3}=\omega (a+bt)\in {\mathcal{E}}_{2}(A), and \overline{E}:=\{({x}_{1},{x}_{2}):({x}_{1},{x}_{2})\in E\}, it is enough to establish that
holds for measurable sets E\subset {\mathbb{R}}^{2}. To do this, we let \mathrm{\Delta}:=\{({t}_{1},{t}_{2}):a<{t}_{1}<{t}_{2}<b\}. The mapping \mathrm{\Phi}:\mathrm{\Delta}\to {\mathbb{R}}^{2} given by \mathrm{\Phi}({t}_{1},{t}_{2})=\gamma ({t}_{2})\gamma ({t}_{1}) is onetoone and the absolute value of the Jacobian determinant J({t}_{1},{t}_{2}) of Φ is given by
Given measurable \mathrm{\Omega}\subset \mathrm{\Delta} and {t}_{2}\in (a,b), we apply Lemma 3.2 with
to obtain
where \tilde{\omega}({t}_{1};{t}_{2}):={max}_{[{t}_{1},{t}_{2}]}\omega. From this, we get
and so
Multiplying by \omega ({t}_{2}) and integrating with respect to {t}_{2} provides us with
Notice that for a<{t}_{1}<{t}_{2}<b, there exists {t}_{\ast}\in [{t}_{1},{t}_{2}] such that \tilde{\omega}({t}_{1};{t}_{2})=\omega ({t}_{\ast}). By Lemma 3.1, we have
which further implies
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). □
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Acknowledgements
The author is grateful to the anonymous referee for valuable comments. This paper was completed with Ajou University research fellowship of 2013.
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Choi, Y. Uniform Lorentz norm estimates for convolution operators. J Inequal Appl 2014, 256 (2014). https://doi.org/10.1186/1029242X2014256
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DOI: https://doi.org/10.1186/1029242X2014256