Endpoint restriction estimates for the paraboloid over finite fields

We prove certain endpoint restriction estimates for the paraboloid over finite fields in three and higher dimensions. Working in the bilinear setting, we are able to pass from estimates for characteristic functions to estimates for general functions while avoiding the extra logarithmic power of the field size which is introduced by the dyadic pigeonhole approach. This allows us to remove logarithmic factors from the estimates obtained by Mockenhaupt and Tao in three dimensions and those obtained by Iosevich and Koh in higher dimensions.


Introduction
Let S denote a hypersurface in R n with measure dσ. The restriction problem for S is to determine for which pairs of (p, q) does there exist an inequality of the form We note that the left-hand side is not necessarily well-defined since we have restricted the functionf to the hypersurface S, a set of measure zero in R n . However, if we can establish this inequality for all Schwartz functions f , then the operator that restrictsf to S can be defined whenever f ∈ L q . In the Euclidean setting, the restriction problem has been extensively studied for many surfaces. In particular, it has been observed that restriction estimates are intimately connected to questions about certain partial differential equations as well as problems in geometric measure theory such as the Kakeya conjecture. The restriction conjecture states sufficient conditions on (p, q) for the above inequality to hold. In the case of the sphere and paraboloid, the question is open in dimensions three and higher. For a survey of restriction results in the Euclidean setting, see [6]. In [5], Mockenhaupt and Tao initiated the study of the restriction phenomena in the finite field setting. This is motivated by both the similarities and the differences between the Euclidean and finite field settings, which suggest that studying restriction phenomena in finite fields may yield insights which are portable to the Euclidean setting, but also that the problems in the finite field setting present unique and independently interesting challenges. In addition, these problems in the finite field setting are closely related to other areas of mathematics, and particularly seem amenable to the use of combinatorial techniques. We now introduce some notation to formally define the problem. We let F denote a finite field of characteristic p > 2. We let S 1 denote the unit circle in C and define e : F → S 1 to be a non-principal character of F . For example, when F = Z/pZ, we can set e(x) := e 2πix/p . We will be considering the vector space F n and its dual space F n * . Following the conventions of [5], we think of F n as endowed with the counting measure dx which assigns mass 1 to each point and F n * as endowed with the normalized counting measure dξ which assigns mass |F | −n to each point (where |F | denotes the size of F , so the total mass is equal to 1 here). To be clear in our calculations, we will always include the appropriate powers of |F | explicitly.
For a complex-valued function f on F n , we define its Fourier transformf on F n * by: For a complex-valued function g on F n * , we define its inverse Fourier transform g ∨ on F n by: It is easy to verify that (f ) ∨ = f and (g ∨ ) = g.
We define the paraboloid P ⊂ F n * as: }. This is endowed with the normalized "surface measure" dσ which assigns mass |P| −1 to each point in P. We note that |P| = |F | n−1 . For a function f : P → C, we define the function (f dσ) ∨ : F n → C as follows: For a complex-valued function f on F n and q ∈ [1, ∞), we define q . (These are the standard definitions of the L q norms, and hence they satisfy the usual properties of norms.) Now we define a restriction inequality to be an inequality of the form where R(p → q) denotes the best constant such that the above inequality holds. Here p ′ and q ′ denote the conjugate exponents of p and q respectively (i.e. 1 p + 1 p ′ = 1). By duality, this is equivalent to the following extension estimate: We will only be considering the case of S = P. We will use the notation X ≪ Y to denote that quantity X is at most a constant times quantity Y , where this constant may depend on the dimension n but not on the field size, |F |. For a finite field F , the constant R(p → q) will always be finite. The restriction problem in this setting is to determine for which (p, q) can we upper bound R(p → q) independently of |F | (i.e. for which (p, q) does R(p → q) ≪ 1 hold).
Mockenhaupt and Tao [5] solved this problem for the paraboloid in two dimensions. In three dimensions, we require −1 not be a square in F . For such F , they showed that R(8/5+ǫ → 4) ≪ 1 and R(2 → 18 5 +ǫ) ≪ 1 for every ǫ > 0. When ǫ = 0, their bounds were polylogarithmic in |F |. Mockenhaupt and Tao's argument for the R(8/5 → 4) estimate proceeded by first establishing the estimate for characteristic functions. Here one can expand the L 4 norm and reduce the problem to combinatorial estimates. A well-known dyadic pigeonhole argument then allows one to pass back to general functions at the expense of a logarithmic power of |F |. The work of Iosevich and Koh in [2], [3], and [4] follows the same approach: first proving restriction estimates for characteristic functions in the finite field setting, and then incurring an extra logarithmic power of |F | in the general estimates obtained through the dyadic pigeonhole argument.
We introduce a method for obtaining general estimates which avoids the polylogarithmic cost of the dyadic pigeonhole technique. Our argument begins by rewriting the L 4 norm as We then adapt the arguments of [5] and [2] to the bilinear variant ||(f dσ) ∨ (gdσ) ∨ || 1/2 L 2 in the case that f and g are characteristic functions. The key point is that we allow f and g to be different characteristic functions -this is what makes our method more powerful than the standard dyadic pigeonhole technique.
To obtain estimates for arbitrary functions f , we can assume that f is non-negative realvalued and decompose f as a linear combination of characteristic functions, where the coefficients are negative powers of two (we can do this without loss of generality by adjusting only the constant of our bound). We can then employ the triangle inequality to upper bound where χ j and χ k are characteristic functions, weighted by negative powers of two. We then apply our bilinear estimate for characteristic functions to these inner terms and use standard bounds on sums to obtain the final estimates.
Our method yields the following theorems: This improves upon Proposition 5.2 in [5] by removing the logarithmic power of |F |. While our argument is elementary and does not use multilinear interpolation, the proof is certainly in the spirit of real interpolation. The second estimate, R(2 → 18 5 ) ≪ 1, follows from the first using the machinery of [5] without modification. After discovering our proof, we learned that in unpublished work Bennett, Carbery, Garrigos, and Wright [1] have independently obtained the end-point results in the 3-dimensional case. Their argument proceeds rather differently than ours and it is unclear if their argument can be extended to the higher dimensional settings. In higher dimensions, we prove: For the paraboloid in n dimensions when n ≥ 4 is even or when n is odd and |F | = q m for a prime q congruent to 3 modulo 4 such that m(n − 1) is not a multiple of 4, we have R( 4n 3n−2 → 4) ≪ 1 and R(2 → 2n 2 n 2 −2n+2 ) ≪ 1. This improves upon Theorems 1, 2, and 3 of [2] by removing the logarithmic power of |F |. We will only prove R( 4n 3n−2 → 4) ≪ 1 here. The estimate R(2 → 2n 2 n 2 −2n+2 ) ≪ 1 follows from the previous estimate from the arguments of [2].
We have restricted our attention to the case of the parabaloid, however our methods are more generic and likely can be combined with the arguments of [3] and [4] for the cases of spheres and more general quadratic surfaces, respectively.

A Restriction Theorem for the Paraboloid in F 3 *
We first prove Theorem 1 for the paraboloid in F 3 * , restated below in an equivalent formulation: For every function f : P → C, we have that: for some constant C.
More concretely, we show below that C = 4 We start by following the strategy of [5], generalizing to the bilinear setting. We employ the following two lemmas. The first one is standard.
Lemma 3. Let P be a collection of points in F 2 , and let L be a collection of lines in F 2 . Then: Lemma 4. We let A, B ⊆ P be arbitrary subsets of P. We define χ A , χ B to be the corresponding characteristic functions from P to {0, 1}. Then: Proof. By definition of the L 2 norm, we have: Using the definitions of (χ A dσ) ∨ and (χ B dσ) ∨ , we can expand this as: We can rewrite this as: For any complex number z, we note that |z| 2 = zz, where z denotes the complex conjugate of z. This allows us to express the above quantity as: Here, we have used that Since these are finite sums, we can interchange their order to obtain: This inner sum will be equal to zero except when a + b = c + d. When this occurs, the inner sum will equal |F | 3 . Thus, we get: Using the fact that A ⊆ P, we observe: This can be bounded above by: We now consider the quantity inside the maximum for an arbitrary, fixed b ∈ P. To bound this, we will use the Galilean transformation g δ : P → P, which is defined for each δ ∈ F 2 * by: We note that for each δ ∈ F 2 * , this is a bijective map from P to itself.
Claim 5. We write b ∈ P as b = (ν, ν · ν), for ν ∈ F 2 * . We also define A ′ := g −ν (A) and B ′ := g −ν (B). We then have: Proof of Claim. We first observe that We will show that for a ∈ A, d ∈ B, a − d + b ∈ P if and only if g −ν (a) − g −ν (d) ∈ P. We can write a as (α, α · α) for some α ∈ F 2 * , and d as (η, η · η) for some η ∈ F 2 * . We can then compute g −ν (a) − g −ν (d) as: This will be an element of P if and only if: which is equivalent to: Now, a − d + b ∈ P holds if and only if: which is also equivalent to: Thus, a − d + b ∈ P if and only if g −ν (a) − g −ν (d) ∈ P. We may then conclude that since g −ν is a bijection from P to P.
We are now interested in bounding the quantity We note that the contribution to this sum from terms where d = 0 is at most |A ′ | = |A|. We note there can be no contribution from terms where a = 0 and d = 0, since having both of d, −d in P is impossible for d = 0. Hence, we have: We now define the sets For each y ∈ F 2 * , we can define a line in F 2 * by ℓ(y) := {x ∈ F 2 * : y · x = y · y}. We now prove that these lines are distinct, i.e. y and ℓ(y) are in bijective correspondence. We suppose that ℓ(y) = ℓ(y ′ ) for y, y ′ ∈ F 2 * . We note that y ∈ ℓ(y) and y ′ ∈ ℓ(y ′ ). Since these lines are the same, we must also have y ∈ ℓ(y ′ ) and y ′ ∈ ℓ(y). By definition of ℓ(y), ℓ(y ′ ), this implies that y · y = y ′ · y = y ′ · y ′ . Hence, (y − y ′ ) · (y − y ′ ) = y · y − 2y ′ · y + y ′ · y ′ = 0. However, since −1 is not a square in F , this implies that y − y ′ must be the zero vector in F 2 * . Thus, y = y ′ . We define L B ′ to be the collection of lines L B ′ := {ℓ(y) : y ∈ X B ′ }. Since these lines are distinct and a ′ − d ′ ∈ P if and only if the corresponding x, y satisfy x ∈ ℓ(y), we have that: We now apply Lemma 3 to conclude: Recalling that This concludes the proof of the lemma.
Proof of Theorem. Our task reduces to proving: We note that if we replace f by the non-negative, real-valued function |f |, then the quantity a+b=c+d a,b,c,d∈P f (a)f (b)f (c)f (d) cannot decrease (by the triangle inequality), and ||f || L 8/5 (P,dσ) remains the same. Therefore, we can assume without loss of generality that f is a non-negative, real-valued function. Moreover, if we replace |f (ξ)| by the smallest power of 2 larger than |f (ξ)|, so that f is a dyadic step-function, the left-hand side will not decrease, while the right-hand side will increase by at most a factor of 2. Thus if we can establish inequality (4) for dyadic step functions for C ′ , the same inequality will hold for all complex-valued functions with C = 2C ′ ).
By the homogeneity of each side of (4), we may assume that ξ∈P |f (ξ)| 8/5 5 8 = 1, and from the previous remarks we may assume that where the E j 's are disjoint subsets of P. We use later the simple consequences that ∞ j=0 2 −j·8/5 |E j | = 1 and |E j | ≤ 2 j·8/5 for all j. It therefore suffices to show that: We calculate From Lemma 4, we have: Using the fact that Thus it suffices to show We consider the two sums 0≤k≤j 2 −j−k |E j | 1/4 |E k | and 0≤k≤j 2 −j−k |E j | 1/2 |E k | 1/2 separately. First we observe: We can alternatively express this last quantity as: Recalling that ∞ k=0 2 −8/5·k |E k | = 1, we have: To bound the other sum, we simply use |E j | ≤ 2 j·8/5 and |E k | ≤ 2 k·8/5 : This shows that for f of the form . Therefore, for arbitrary complexvalued functions f , we have ||(f dσ) ∨ || L 4 (F 3 ,dx) ≤ C||f || L 8/5 (P,dσ) , where C can be set to . This concludes the proof of the theorem.

Restriction Theorem for the Paraboloid in Higher Dimensions
We now consider the paraboloid P in F n for values of n > 3. We begin by presenting a combinatorial lemma.

A Combinatorial Lemma
Lemma 6. For any sets A, B ⊆ P ⊆ F n where n ≥ 4 is even or n is odd and |F | = q m for some prime q congruent to 3 modulo 4 with m(n − 1) not a multiple of 4, we have: Proof. We follow the strategy used to prove Lemmas 7 and 8 in [2], generalizing it appropriately to allow arbitrary A and B (in the [2] lemmas, A = B). We first note: We can express a point a ∈ A as a = (a, a · a), for some a ∈ F n−1 * .
We let δ denote the function on F * which is 1 when the input is 0 and is 0 otherwise. We then have: Now, for any value t ∈ F * , we note that δ(t) = |F | −1 s∈F * e(st). Thus, we have: We note that when s = 0, the value of the sum over a, b, d is equal to |A||B| 2 . This gives us: e(s(a·b−a·d−b·d+d·d)).
We now consider upper bounding the quantity By the triangle inequality, this is: By applying the Cauchy-Schwarz inequality to the sum over a, this is: Again employing the triangle inequality, we have: By applying the Cauchy-Schwarz inequality to the sum over d, this is: Since B ⊆ P, this is: For any a ∈ F n−1 * , we define the quantity M (a) (with respect to B) as in [2]: In [2], they prove for even n ≥ 4 that 1 holds for all a ∈ F n−1 * . This gives us: 1 Actually, they state this for a such that a ∈ B (since A = B in their case), but their proof never uses that a ∈ B, so it extends without modification to all a's.
By substituting this into our bounds above, we have that: For odd n when |F | = q m for some prime q congruent to 3 modulo 4 with m(n − 1) not a multiple of 4, they prove in [2] that 2 M (a) ≪ |F | n |B| + |F | By substituting this into our bounds above, we have that: We note that this is actually a somewhat better estimate than the lemma requires, since |F |

Proof of Theorem 2
We recall that p := 4n 3n−2 . We now prove Theorem 2, restated below in an equivalent formulation: Theorem 2. When n ≥ 4 is even or when n is odd and |F | = q m for a prime q congruent to 3 modulo 4 such that m(n − 1) is not a multiple of 4, for every function f : P → C we have that: Proof. Expanding the L 4 norm, we see that our task reduces to proving: We will find it more convenient to prove the equivalent formulation: .
Hence, we can rewrite the inequality to be proved as: Lemma 6 gives us an upper bound on the quantity ( a+b=c+d a,c∈E j b,d∈E k 1) 1/2 , but we can also obtain the simpler upper bound of |E j ||E k | 1/2 by noting that for fixed values of a, c ∈ E j , b ∈ E k , there is at most one value of d ∈ E k which satisfies a + b = c + d.
To bound the sum for values of j > U , we use that |E j | ≤ |F | n−1 = |P| and |E k | ≤ 2 kp : The geometric sum over k is convergent, so we have ≪ |F | −1/2+n−1 j≥U 2 j(p/2−2) . The geometric sum over j is now also convergent, so we have: This concludes the proof of the theorem.

Acknowledgments
We would like to thank Jeff Vaaler for helpful discussions.