Uniform Lorentz norm estimates for convolution operators
© Choi; licensee Springer 2014
Received: 8 January 2014
Accepted: 1 July 2014
Published: 22 July 2014
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.
for . Here and in what follows, we denote . Curves of the form are said to be simple according to Drury and Marshall . The measure supported on the curve is known as the affine arclength measure, which is based on the affine arclength parameter as in , and was introduced by Drury and Marshall  in dealing with the Fourier restriction problem related to curves, and later by Drury  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 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 of is contained in the convex hull of and uniform estimates in a, b, and ϕ are expected only for . Many conditions to guarantee optimal uniform - estimates have been known so far. See [3, 5–12] for example. Among other things, the author proved the following.
Theorem 1.1 (Choi )
holds uniformly in .
Under somewhat stronger assumptions on or , the endpoint Lebesgue norm estimate aforementioned can be improved to optimal Lorentz norm estimates, namely from into and into . 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 )
for all , where C is a constant depending only on A.
For the proof of the optimality, see  by Stovall along with  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 , the author aims to establish a uniform optimal Lorentz norm improving estimate under a condition on averages of the square of . 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 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 , , was introduced by Bak et al.  in studying Fourier restriction estimates related to degenerate curves.
Definition 2.2 For an interval J and a positive real number A, a function is said to be a member of if
Φ is monotone; and
whenever and ,
The condition (1.2) can be rewritten as .
It is a simple matter to check:
for all ;
if and only if ;
implies for all ; and
implies for all .
If , , and , then by Hölder’s inequality.
The class is essentially the class of logarithmically concave functions, which already encompasses many useful examples. Simplest examples are the exponential function and , , for . More interesting example is the function , , which models a curve ‘flat’ at the origin. A hierarchy of flatter functions that belong to was constructed by Bak et al. .
For a polynomial of degree N, belongs to after (possibly) decomposing the real line into at most intervals.
Nevertheless, there are functions that belong to but do not belong to for any . Two examples of curves that our result covers that are not covered in  can be constructed with the aid of the examples given below.
which implies . In view of Remark 2.3, given , if . One can easily see for any and .
Example 2.5 Consider given by . Then we have and as , which clearly implies for all . On the other hand, by the following.
Proposition 2.6 Let . Suppose that for some . Then the function Φ given by belongs to for .
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 .
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 is strictly stronger than the condition in Theorem 2.7, and therefore our result improves Theorem 1.2.
An explicit example is also available. Consider defined for , where c is a large constant. A simple calculation shows . By Proposition 2.6, for some . 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.
3 Proof of the main theorem
Before we prove the theorem, we begin with a couple of lemmas.
whenever and .
By taking the th power, we obtain the desired estimate. □
The following lemma, which is nearly a triviality, generalizes a version of Lemma 2.2 in .
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, , 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 , we denote the Lebesgue measure and the characteristic function of E by and , respectively. We also write .
Letting and making a change of variables, we obtain the desired estimate (3.2). □
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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