Sharp endpoint estimates for the $X$-ray transform and the Radon transform in finite fields

This note establishes sharp $L^p-L^r$ estimates for $X$-ray transforms and Radon transforms in finite fields.


Introduction
Let F d q , d ≥ 2, be a d-dimensional vector space over the finite field F q with q elements. We endow F d q with a normalized counting measure dx. For each q, we denote by M q a collection of certain subsets of F d q . Recall that a normalized surface measure dσ on M q can be defined by the relation where |M q | denotes the cardinality of M q and Ω is a complex-valued function on M q . For any complex-valued function f on (F d q , dx) and w ∈ M q , we consider an operator T Mq defined by We are interested in determining exponents 1 ≤ p, r ≤ ∞ such that the following inequality holds: T Mq f L r (Mq,dσ) f L p (F d q ,dx) for all f on F d q , where the operator norm of T Mq is independent of q, the size of the underlying finite field F q . If M q is Π k , a collection of all k-planes in F d q with 1 ≤ k ≤ d − 1, then the operator T Π k is called as the k-plane transform. In particular, T Π 1 and T Π (d−1) are known as the X-ray transform and the Radon transform respectively. In Euclidean space, the complete mapping properties of k-plane transforms were proved by M. Christ in [1]. Readers may refer to [9], [8], [3] for the description of k-plane transforms in the Euclidean setting. In 2008, Carbery, Stones, and Wright [2] initially studied the mapping properties of k-plane transforms in finite fields. Using combinatorial arguments, they proved the following theorem.  T holds with C independent of |F q |, then (1/p, 1/r) lies in the convex hull H of ((k + 1)/(d + 1), 1/(d + 1)), (0, 0), (1, 1) and (0, 1).
Notice from Theorem 1.1 that if one could show that the restricted type inequality (1.2) can be replaced by the strong type inequality, then the mapping properties of the k-plane transforms in finite fields would be completely established. Namely, our task would prove the following conjecture.
where C is independent of |F q |.
1.1. Statement of main results. In this paper we prove that Conjecture 1.2 is true for the X-ray transform and the Radon transform. More precisely, we obtain the following theorem.
where C is independent of |F q |.
In order to prove Theorem 1.3 for the X-ray transform (k = 1), we shall adapt both the combinatorial arguments in [2] and the skills in [6] for endpoint estimates. On the other hand, a Fourier analytic argument will be required to prove Theorem 1.3 for the Radon transform (k = d−1) . Remark 1.4. After writing this paper, the author realized that our result for the X-ray transform is a corollary of Theorem 1.1 in the paper [4]. This was pointed out by R. Oberlin.

Proof of the mapping properties of the X-ray transform
In this section, we restate and prove Theorem 1.3 in the case of the X-ray transform. Namely, we prove the following statement which implies the sharp boundedness of the X-ray transform.
where C is independent of |F q |.
Proof. We begin by following the argument in [6]. Without loss of generality, we may assume that f is a non-negative real-valued function and Thus, it is natural to assume that f ∞ ≤ 1. Furthermore, we may assume that f is written by a step function where the sets E i are disjoint subsets of F d q , and here, and throughout the paper, we write E(x) for the characteristic function on a set E ⊂ F d q . From (2.1) and (2.2), we also assume that for all j = 0, 1, · · · .
Since dx is the normalized counting measure on F d q , the assumption (2.1) shows that we only need to prove where f satisfies (2.2) and (2.3). Since we have assumed that f ≥ 0, it is clear that T Π 1 f is also a non-negative real-valued function on Π 1 . By expanding the left-hand side of (2.4) and using the facts that |w| = q for w ∈ Π 1 and |Π 1 | ∼ q 2(d−1) , we see that where the last line follows from the symmetry of i 0 , · · · , i d . We now follows the argument in [2]. Notice that we can write denotes the smallest affine subspace containing the elements x 0 , . . . , x d . In addition, observe that if s > 1 and (x 0 , . . . , x d ) ∈ ∆(s, i 0 , . . . , i d ), then the sum over w ∈ Π 1 vanishes. On the other hand, if s = 0, 1, then the sum over w ∈ Π 1 is same as the number of lines containing the unique s-plane, that is ∼ q (d−1)(1−s) . From these observations and (2.4), it is enough to prove that for all Namely, it suffices to prove that for every d ≥ 2 and s = 0, 1, where ∆(s, i 0 , . . . , i d ) is defined as before, and the sets E i , i=0,1,. . . , satisfy (2.3). Suppose that Since the sum of a convergent geometric series is similar to the value of the first term, we have the desirable conclusion for s = 0: where the last equality is obtained from (2.3).
Next, we assume that s = 1. We must show that for all E i , i = 0, 1, . . . , satisfying (2.3), we have , then all points x i 0 , . . . , x i d must lie on a line, which is determined by at least two different points of them . Therefore, for each l = 1, 2, . . . , d,, we can define where we recall that [α 1 , . . . , α s ] means the smallest affine subspace containing all points By the definition of L(l), l = 1, . . . , d, it follows that for every l = 1, . . . , d, To see this, first fix Since |E i l | ≤ q d and l ≥ 1, it is easy to see that |E i l | (l−1)/d q 1−l 1. Therefore, it follows that Compute the inner summations by checking that each of them is a convergent geometric series. It follows that where the last equality follows from (2.3). Thus, we complete the proof of Theorem 2.1.
Remark 2.2. It seems that the similar arguments as above work for settling Conjecture 1.2, but it may not be simple to estimate |∆(s, i 0 , . . . , i d )|.

Proof of mapping properties of the Radon transform
In this section, we prove Theorem 1.3 in the case of the Radon transform. Namely, we shall prove the following.
Theorem 3.1. Let d ≥ 2 be any integer. Then, where C is independent of |F q |.
Proof. First, notice that if the dimension d is two, then the statement of Theorem 3.1 follows immediately from Theorem 2.1. We therefore assume that d ≥ 3. As before, we may assume that f is a non-negative real function and Moreover, we may assume that the function f is a step function: where the sets E i are disjoint subsets of F d q . Notice that (3.1) and (3.2) imply that On the other hand, for a fixed w ∈ Θ, there is a unique line passing through the origin, say L w , such that w = {x ∈ F d q : w ′ · x = 0 for all w ′ ∈ L w \ {(0, . . . , 0}}. By selecting one specific w ′ ∈ L w \ {(0, . . . , 0)} we can identify w ∈ Θ with the specific point w ′ ∈ L w \ {(0, . . . , 0)}. Throughout the paper, we denote by S the collection of the specific points each of which is chosen from L w \ {(0, . . . , 0)} for every w ∈ Θ. 1 Thus, we also assume that if w ∈ Θ, then 1 In the Euclidean setting, one can consider the set S as a half part of the unit sphere. However, it is not true in general in the finite field setting. For example, if the dimension d is four and −1 ∈ Fq is a square number, then the line l = {t(i, 1, i, 1) : t ∈ Fq} does not intersect the set {x ∈ F 4 q : x 2 1 + · · · + x 2 4 = 1}.
where w ′ ∈ S. Since Π d−1 = H ∪ Θ and H ∩ Θ = ∅, the Radon transform T Π d−1 can be viewed as where the operators T 0 and T 1 are defined as and In order to prove Theorem 3.1, it therefore suffices to show that the following two inequalities hold: , and (3.5) , where the functions f satisfy (3.1), (3.2), and (3.3).

Proof of the inequality (3.4).
Let us denote by χ the canonical additive character of F q (see [7] or [5]). Recall that the orthogonality relation of χ holds: Using the orthogonality relation of χ, we have where we used the facts that dx and dσ are the normalized counting measure on F d q and the normalized surface measure on Π d−1 respectively. To prove the inequality (3.4), it remains to prove that for all functions f satisfying (3.1), (3.2), and (3.3), where the last equality follows from (3.1). We need the following lemma.
Lemma 3.2. Let d ≥ 3. Then, for every subset E ⊂ F d q , we have Proof. Since d ≥ 3, we see that the statement of Lemma 3.2 follows immediately by interpolating the following two estimates: for all indicator functions E(x) on F d q , It remains to prove that (3.8) holds. It follows that for every set E ⊂ F d q , where we used the fact that S can be identified with Θ. Using a change of variables, we see that for each w ′ ∈ S, Γ(w ′ ) = Γ(tw ′ ) for all t ∈ F * q . By the definition of S, it therefore follows that Since |Π d−1 | ∼ q d , if we expand the square term and apply the orthogonality relation of χ to the sum over w ′ ∈ F d q , then we see that Thus, the proof of Lemma 3.2 is complete.
We now prove (3.6). Since we have assumed that f is considered as a step function (3.2), it follows that where the last line follows from the symmetry of i, j. By Hölder's inequality, the inequality (3.7), and Lemma 3.2, (3.6) will follow if we prove that 1.
This can be justified by making use of (3.3) and computing the summation over j variable: which completes the proof of (3.4).

3.2.
Proof of the inequality (3.5). By showing that the inequality (3.5) holds, we shall complete the proof of Theorem 3.1. We shall take the same steps as in the previous subsection. From the orthogonality relation of χ, it follows that for all w ∈ Π d−1 , As before, it is easy to see that .
Thus, it is enough to prove that for all functions f satisfying (3.1), (3.2), and (3.3), where the last equality follows from (3.1). From the same arguments as in the proof of (3.4), our task is only to obtain Lemma 3.2 for the operator T ⋆⋆ 1 . As in the proof of Lemma 3.2, it suffices to prove the following two equalities: for every subset E of F d q , The inequality (3.9) follows immediately from the same argument as before. To prove (3.10), we observe that Recall that H ⊂ Π d−1 can be identified with F d q \ {(0, . . . , 0)}. Thus, if we dominate the sum over w ∈ H by the sum over w ′ ∈ F d q , expand the square term, and use the orthogonality relation of χ over the variable w ′ ∈ F d q , then it follows that χ(−t(1 − u)).