Existence of Extremizers for a Model Convolution Operator

Abstract

The operator T , defined by convolution with the affine arc length measure on the moment curve parametrized by h(t) = (t, t 2 , ..., t d ) is a bounded operator from L p to L q if ( 1 p , 1 q ) lies on a line segment. In this article we prove that at non-end points there exist functions which extremize the associated inequality and any extremizing sequence is pre compact modulo the action of the symmetry of T . We also establish a relation between extremizers for T at the end points and the extremizers of an X-ray transform restricted to directions along the moment curve. Our proof is based on the ideas of Michael Christ on convolution with the surface measure on the paraboloid.The above theorem was proved for d = 2 by Littman in [20]. Oberlin [21] showed that it is sufficient to satisfy the above condition when d = 3. The theorem was proved up to the end point for any dimension by Christ in [2]. Extending ideas from [2], Stovall proved the strong type end point bound [22].By using methods based on several observations and results of Christ (see [4,2,3,22]), together with some refinements by Stovall and Dendrinos [12] we have been able to refine the associated inequality for this operator. To state our results, first we need to introduce some definitions.Let (p, q) be as above. Let A be the operator norm of T . That isDefinition 1.4. δ-Quasiextremizer pair. Let δ>0. We say that an ordered pair (f, g) of measurable functions on R d is an δ-quasiextremizer pair ifDefinition 1.5. Extremizing sequence. An extremizing sequence is any sequence f n ∈ L p such that f n L p = 1, T f n L q → A.Let us denote the moment curve by h(t) = (t, t 2 , ..., t d ). Our main theorems are as follows.

Compactness of a restricted X-ray transform

“…We are finally ready to prove Lemma 3.2. This follows easily from the argument for a similar result in Lemma 11.2 in [1]. For the sake of completeness we give a very brief sketch of the argument here.…”

“…The proof of this observation is similar to that of Lemma 5.6 in [7] and also to part of the proof of Lemma 7.2 in [1], so we omit the details.…”

“…Proof. The following proof is adaptation of an argument which originated in [4] and used in similar arguments in [1]. We give the proof for the case when d = 2D+1 ≥ 3 is odd, for the other case being identical.…”

“…This mock-distance has the following immediate properties. For the elementary proof of these properties we direct the reader to a similar proof in section 5 of [1].…”

“…Our motivation for investigating the compactness of the X-ray transform originated partly due to the results in [1]. There the author was examining similar questions for a different operator known as an averaging operator or generalized Radon transform in recent literature.…”