Extension and averaging operators in vector spacesover finite fields

Doowon Koh and Chun-Yen Shen

Abstract. We study the mapping properties of both extension operators and averagingoperators associated with the algebraic variety S = {x ∈ Fd

q : Q(x) = 0} where Q(x) isa nondegenerate quadratic form over the finite field Fq. Carbery, Stones, and Wright [3]established a method to deduce Lp − Lr estimates for averaging operators over a generalvariety. As a core result in this paper, we give the sharp estimates, except two endpoints, ofthe averaging operator over S in even dimensions, which could not be obtained directly fromthe method by aforementioned authors. The key idea is to obtain more delicate L2 averagingestimates, which are related to the dual extension estimates associated with S. In threedimension, we also study extension problems related to varieties H = {x ∈ F3

q : P (x) = 0}where P ∈ Fq[x1, x2, x3] is arbitrary homogeneous polynomials of degree � ≥ 2. We showthat if H ⊂ F3

q does not contain a plane, then the homogeneous variety H also yieldsthe sharp exponents as conical extension estimates due to Mockenhaupt and Tao [14]. Inaddition, we indicate that if the dimension d ≥ 3 is odd, then the uniform Fourier decayon S ⊂ Fd

q enables us to easily deduce sharp Lp − Lr averaging in all odd dimensions andextend the cone extension results in three dimension to higher dimensions.

Contents

1. Introduction and statement of results 12. Discrete Fourier analysis 83. Key lemmas and their proofs 84. Proof of Theorem 1.4 (main result on averaging problems) 125. The proof of Theorem 1.8(main result on extension problems) 146. Appendix 15References 17

1. Introduction and statement of results

In recent years some of the most challenging problems in Euclidean harmonic analysishave been intensively studied in the finite field settings, in part the vector spaces over finitefields serve a nice model and possess some very interesting phenomena. In addition, somedeep number theoretic arguments also play an important role in studying these peoblems. Inthis paper we mainly investigate the mapping properties of extension operators and averaging

2000 Mathematics Subject Classification. 42B05, 11T24, 52C10.1

operators associated with algebraic surfaces in vector spaces over finite fields. Let Fdq , d ≥ 2,be a d-dimensional vector space over a finite field Fq with q elements. To avoid a degeneratecase, we assume that the characteristic of Fq is large enough. We consider an algebraicvariety V ⊂ Fdq with ∣V ∣ ∼ qd−1, where the implicit constants in ∼ are independent of thesize of the underlying finite field Fq. The variety V is (in principle) a (d − 1)-dimensionalhypersurface in Fdq . We put normalized counting measure d� on V. One of the problemsconsidered in this paper is the existence of averaging estimates of the form

(1.1) ∥f ∗ d�∥Lr(Fdq ,dx) ≤ Cd,p,r∥f∥Lp(Fdq ,dx)

where the Lebesgue norms are taken with respect to normalized counting measure on Fdq andthe constant is supposed to be independent of f and q, the size of the underlying finite fieldFq. We shall denote by A(p → r) the best constant satisfying the averaging estimate (1.1).We are interested in determining the exponents 1 ≤ p, r ≤ ∞ such that A(p → r) ≲ 1,here, throughout the paper, the implicit constant in ≲ may depend on d, p, r and the degreeof V but is independent of q. This problem is called the averaging problem in the finitefield setting which was initially investigated by Carbery, Stones, and Wright [3] for certainvarieties. Moreover, this problem has also been studied in the Euclidean spaces, and manydeep results were already established (see [15], [16], [10], and [8]).

It is known from [3] that the necessary condition for A(p→ r) ≲ 1 is

(1.2)

(1

p,1

r

)∈ Ω0,

where Ω0 denotes the convex hull of points (0, 0), (0, 1), (1, 1), and (d/(d + 1), 1/(d + 1)).However, if the dimension d is even and V contains a d/2-dimensional affine subspace Γ ,then the necessary condition (1.2) can be improved by the conditions

(1.3)

(1

p,1

r

)∈ Ω1,

where Ω1 is defined as the convex hull of points (0, 0), (0, 1), (1, 1),

P1 =

(d2 − 2d+ 2

d(d− 1),

1

(d− 1)

)and P2 =

(d− 2

d− 1,d− 2

d(d− 1)

).

In fact, the improved necessary conditions are obtained by testing the inequality (1.1) withthe indicator function on the d/2-dimensional affine subspace Γ. The authors in [3] alsointroduced a simple method to deduce averaging estimates from size estimates on (d�)∨. Ifwe follows the proof of Theorem 3 in [3], then the following lemma can easily be reduced.

Lemma 1.1. Let d� be the surface measure on the algebraic variety V ⊂ Fdq with ∣V ∣ ∼ qd−1.

If ∣(d�)∨(m)∣ ≲ q−�2 for all m ∈ Fdq ∖ (0, . . . , 0) and for some � > 0, then we have

A(p→ r) ≲ 1 whenever

(1

p,1

r

)∈ Ω(�),

where Ω(�) is the convex hull of points (0, 0), (0, 1), (1, 1), and ((� + 1)/(� + 2), 1/(� + 2)) .

As a consequence of a deep algebraic geometric method, the following lemma is well-known.

2

Lemma 1.2 ([2]). Let P (x) =d∑j=1

ajxsj ∈ Fq[x1, . . . , xd] with s ≥ 2, aj ∕= 0 for all j = 1, . . . , d.

For each t ∈ Fq, define Ht = {x ∈ Fdq : P (x) = t}. Suppose that the characteristic of Fq issufficiently large so that it does not divide s. Then, we have

∣(d�t)∨(m)∣ ≲ q−d−12 for all m ∈ Fdq ∖ {(0, . . . , 0)}, t ∈ Fq ∖ {0},

and

(1.4) ∣(d�0)∨(m)∣ ≲ q−d−22 for all m ∈ Fdq ∖ {(0, . . . , 0)},

where d�t denotes a normalized surface measure on Ht.

From the necessary condition (1.2), Lemma 1.1, and Lemma 1.2, it is an easy matterto obtain sharp averaging estimates over Ht, t ∕= 0. Thus, an interesting question is raised,namely : the averaging problem on the homogeneous variety H0. Observe that if we applyLemma 1.1 and the estimate (1.4), then the following averaging estimates over H0 can beobtained:

A(p→ r) ≲ 1 if

(1

p,1

r

)∈ Ω3,

where Ω3 is the convex hull of points (0, 0), (0, 1), (1, 1), and ((d− 1)/d, 1/d) . However, theseresults are even worse than the improved necessary conditions (1.3). In this situation, wemay have two natural questions. First, whether the estimates (1.4) are improvable or not ?Second, if the estimates (1.4) were sharp, then how to obtain the sharp averaging estimateson H0? This paper was mainly motivated on these questions. To obtain partial answers tothe questions, we consider an algebraic variety

(1.5) S = {x ∈ Fdq : Q(x) = 0}

where Q(x) ∈ Fq[x1, . . . , xd] is a nondegenerate quadratic form over the finite field Fq. By anonsingular linear substitution, any nondegenerate quadratic form Q(x) ∈ Fq[x1, . . . , xd] canbe transformed into a1x

21 + ⋅ ⋅ ⋅+adx

2 for some aj ∈ Fq ∖{0}, j = 1, . . . , d (see [12]). Withoutloss of generality, we therefore assume that for some aj ∕= 0,

(1.6) S = {x ∈ Fdq : a1x21 + ⋅ ⋅ ⋅+ adx

2d = 0}.

We shall use d� to denote the normalized surface measure on S :∫S

f(x) d�(x) =1

∣S∣∑x∈S

f(x),

here, and throughout the paper, we denote by ∣E∣ the cardinality of the set E ⊂ Fdq . As asimple application of Gauss Sum estimates (see our Lemma 3.1 and Corollary 3.2 below),we shall see that if the dimension d ≥ 3 is odd, then

∣(d�)∨(m)∣ ≲ q−d−12 for all m ∕= (0, . . . , 0),

which improves the estimate (1.4). On the other hand, if the dimension d ≥ 4 is even, then

(1.7) ∣(d�)∨(m)∣ ≲ q−d−22 for all m ∕= (0, . . . , 0),

which is sharp and is the same estimate as in (1.4). Therefore, the following corollary followsimmediately from Lemma 1.1 and the necessary conditions (1.2).

3

Corollary 1.3. Let S ⊂ Fdq , defined as in (1.5) or (1.6). If d ≥ 3 is odd, then the necessaryconditions (1.2) are in fact sufficient conditions. On the other hand, if d ≥ 4 is even, then

(1.8) A(p→ r) ≲ 1 whenever

(1

p,1

r

)∈ Ω3,

where Ω3 denotes the convex hull of points (0, 0), (0, 1), (1, 1), and ((d− 1)/d, 1/d) .

1.1. Statement of main result on averaging operators. Notice that if d ≥ 4 iseven, then the averaging results (1.8) are much weaker than the necessary conditions (1.3)even if we applied Lemma 1.1 with the sharp Fourier estimate (1.7). In addition to Lemma1.1, we shall take an advantage of precise L2 averaging estimates for which we shall usedual extension results. As a consequence, we have the following main theorem for averagingproblems over S, which is sharp except two endpoints in general cases.

Theorem 1.4. Let S ⊂ Fdq be a nondegenerate quadratic surface defined as in (1.5) or (1.6).Suppose that d ≥ 4 is even. Then,

A(p→ r) ≲ 1 if

(1

p,1

r

)∈ Ω ∖ {P1, P2},

where Ω denotes the convex hull of points (0, 0), (0, 1), (1, 1),

P1 =

(d2 − 2d+ 2

d(d− 1),

1

(d− 1)

)and P2 =

(d− 2

d− 1,d− 2

d(d− 1)

).

Moreover, if (1/p, 1/r) = P1 or P2, then

A(p→ r) ≲ log q.

In particular, if S contains a d/2-dimensional subspace and (1/p, 1/r) /∈ Ω, then Lp − Lr

estimate is not possible.

The complete proof of Theorem 1.4 will be given in Section 4.

1.2. Introduction of extension problems. The other topic in this paper is to studyextension problems associated with homogeneous varieties in Fdq . The extension theoremswill be applied to prove Theorem 1.4, the main result of the averaging problem over S ineven dimensions. Mockenhaupt and Tao [14] first set up and studied extension problemsassociated with various algebraic varieties in Fdq . In the finite field setting, extension problemsfor paraboloids and spheres have been well studied in [14], [5],[6], and [11]. However,extension results for homogeneous varieties in Fdq are not known except the cone in threedimension, in part because it is very difficult to estimate the sharp Fourier coefficients of d�.In this paper we also investigate the mapping properties of extension operators associatedwith homogeneous varieties. Let P ∈ Fq[x1, . . . , xd] be a polynomial with degree ≥ 2. Weconsider an algebraic variety

V = {x ∈ Fdq : P (x) = 0},

where we assume that ∣V ∣ ∼ qd−1. If P ∈ Fq[x1, . . . , xd] is a homogeneous polynomial ofdegree � ≥ 2, then we call the variety V as a homogeneous variety of degree � ≥ 2. Inparticular, if P ∈ Fq[x1, . . . , xd] is a nondegenerate quadratic form, then V is called a non-degenerate quadratic variety. In addition, if P ∈ Fq[x1, x2, x3] is a homogeneous polynomialof degree � ≥ 2, and V ⊂ F3

q does not contain any plane, then the variety V is called a4

nondegenerate homogeneous variety of degree � ≥ 2.

Throughout the paper we denote by “dx” or “dy” the normalized counting measure onthe vector space Fdq . On the other hand, we shall use “dm” to indicate the counting measure

on the dual space of Fdq . For a simple notation, we also write the notation “Fdq” for the dual

space, because the meaning of Fdq is clear from the measure notation. We endow the varietyV with the normalized surface measure d�. For each 1 ≤ p, r ≤ ∞, we denote by R∗(p→ r)the smallest constant such that the following extension estimate hold:

∥(fd�)∨∥Lr(Fdq ,dm) ≤ R∗(p→ r)∥f∥Lp(V,d�).

Then, the extension problem in the finite field setting is to determine the exponents 1 ≤p, r ≤ ∞ such that

R∗(p→ r) ≲ 1,

here, we recall that the implicit constant in ≲ may depend on d, p, r, the degree of V and theratio ∣V ∣/qd−1, but is independent of the functions f and q. By duality, we also see that thequantity R∗(p→ r) is also the smallest constant such that the following restriction estimateholds: for every function g on (Fdq , dm),

(1.9) ∥g∥Lp′ (S,d�) ≤ R∗(p→ r)∥g∥Lr′ (Fdq ,dm),

here, throughout the paper, p′ and r′ denote the dual exponents of p and r respectively. Inother words, 1/p+ 1/p′ = 1 and 1/r+ 1/r′ = 1. In the Euclidean space, these problems havebeen well studied, but have not yet been solved in higher dimensions or general setting (see[13], [1], [21], [18], and [19]).

It is well-known from [14] that the necessary conditions for R∗(p→ r) ≲ 1 take

(1.10) r ≥ 2d

d− 1and r ≥ dp

(d− 1)(p− 1).

However, if V contains a k-dimensional affine subspace W (∣W ∣ = qk), the necessary condi-tions (1.10) can be improved to the conditions

(1.11) r ≥ 2d

d− 1and r ≥ p(d− k)

(p− 1)(d− 1− k)

Furthermore, Mockenhaupt and Tao [14] showed that if the variety V is a cone in threedimension, V = {x ∈ F3

q : x21 = x2x3}, then the necessary conditions (1.11) can even be

improved to the conditions

r ≥ 4 and r ≥ 2p

p− 1.

By obtaining the critical L2−L4 estimate for the cone in F3q, they also proved that the neces-

sary conditions are also sufficient. As a key idea to obtain the critical L2−L4 estimate, theyhighly adapted the geometric properties of cones, however it turns out that such approachonly yields somewhat weak results for higher dimensional case.

Now, we address three main reasons why we consider extensions problems related to ho-mogeneous varieties in Fdq . First, we show that the sharp L2 − Lr-extension estimate can beused to prove sharp, strong weak-type averaging estimates in even dimensions (Theorem 1.4),which could not be derived directly from Lemma 1.1. Second, using geometric properties on

5

homogeneous varieties, we prove that the sharp conical extension results in three dimension,due to Mockenhaupt and Tao [14], in fact hold for any nondegenerate homogeneous varietyV ⊂ F3

q with any degree � ≥ 2. This fact is contrasted to well-known properties of extensionoperators in the Euclidean space. Last, we remark that well-established Tomas-Stein argu-ment can be applied to obtain the sharp conical extension results in three dimension andextend such results to higher dimensions.

1.3. Statement of results on extension problems. Mockenhaupt and Tao [14] es-tablished a simple method to deduce L2 − Lr extension estimate from the uniform decayrate of (d�)∨. The method is well known as the Tomas-Stein type argument which is givenin this content as follows.

Lemma 1.5 (Tomas-Stein Type argument in the finite field setting). Let d� be the normal-ized surface measure on the algebraic variety V ⊂ Fdq with ∣V ∣ ∼ qd−1. If ∣(d�)∨(m)∣ ≲ q−

�2

for some � > 0 and for all m ∈ Fdq ∖ (0, . . . , 0), then we have

R∗(

2→ 2(� + 2)

�

)≲ 1.

It is clear from this lemma that the better uniform Fourier decay on a variety yieldsthe better L2 − Lr extension estimates. Since homogeneous varieties are composed of acollection of lines, someone may have a prejudice that uniform Fourier decay rate on ahomogeneous variety such as a cone is worse than a sphere and a paraboloid. However,this is not necessarily true in the finite field setting. In fact, our Lemma 3.1 and Corollary3.2 below illustrate that any nondegenerate quadratic variety in odd dimensions can yield asgood uniform Fourier decay as the sphere or the paraboloid, although it is not the case in evendimensions. From this simple observation, Lemma 1.5 implies the sharp conical extensionresult in F3

q, which was already established from geometric observations by Mockenhaupt andTao [14]. Moreover, one can obtain the extension results for the cone in higher dimensions.More precisely, we have the following.

Theorem 1.6. Let S ⊂ Fdq be a nondegenerate quadratic variety , defined as in (1.5) or(1.6). If d ≥ 3 is odd, then we have

(1.12) R∗(

2→ 2d+ 2

d− 1

)≲ 1.

Thus, it follows from the necessary conditions (1.11) that if S contains a (d−1)/2-dimensionalsubspace M , then (1.12) gives a sharp L2−Lr estimate. On the other hand, if d ≥ 4 is even,then

(1.13) R∗(

2→ 2d

d− 2

)≲ 1.

Therefore, it follows from the necessary conditions (1.11) that if the surface S contains d/2-dimensional subspace Γ, then (1.13) gives a sharp L2 − Lr extension estimate.

Remark 1.7. Since any nondegenerate quadratic surface can be transformed to S = {x ∈Fdq : a1x

2 + ⋅ ⋅ ⋅+ adx2 = 0} with aj ∈ Fq ∖ {0}, j = 1, . . . , d. If one takes aj = 1 for j odd and

aj = −1 for j even, then the surface S in odd dimensions contains a (d− 1)/2-dimensionalsubspace

M ={

(t1, t1, t2, t2, . . . , t(d−1)/2, t(d−1)/2, 0) ∈ Fdq : tk ∈ Fq, k = 1, 2, . . . , (d− 1)/2},

6

and the surface S in even dimensions contains a d/2-dimensional subspace

Γ ={

(t1, t1, t2, t2, . . . , td/2, td/2) ∈ Fdq : tk ∈ Fq, k = 1, 2, . . . , d/2}.

Also notice that if−1 ∈ Fq is a square number, then S always contains a (d−1)/2-dimensionalsubspace for odd dimensions and a d/2-dimensional subspace for even dimensions. The sharpL2 − Lr extension estimate (1.13) in even dimensions shall be applied to prove our mainaveraging result (Theorem 1.4).

We now state the most interesting result on extension problems in this paper. As men-tioned before, the conical L2 − L4 extension estimate in three dimension is possible, whichimplies the complete answer to the conical extension problem in F3

q. We also know fromthe first part of our Theorem 1.6 that the nondegenerate quadratic surface S also yields theL2−L4 extension estimate in three dimension. These two same results may not be surprisingto us, because the orders of both varieties are same, namely two. What makes us interestedis our main theorem below which says that any nondegenerate homogeneous variety H ⊂ F3

q

with any order ≥ 2 yields the L2 − L4 extension estimate. This presents an interesting factthat there exist some differences between the finite field case and the Euclidean case. Forexample, let us consider a set V = {x ∈ ℝ3 : x4

1 + x42 − x4

2 = 0}. In the Euclidean case, it iswell known that the extension estimate for a compact subset of V is much worse than thatfor the compact subset of U = {x ∈ ℝ3 : x2

3 = x21 + x2

2} (see [4] or the pages 414 and 418 in[17]). However, it seems that the degree of varieties does not affect on extension estimates inthe finite field setting. Our main theorem for extension problems is as follows and geometricobservations make a crucial role to prove it. We shall give the proof in Section 5.

Theorem 1.8. Given a homogeneous polynomial P ∈ Fq[x1, x2, x3] with degree � ≥ 2,consider the homogeneous variety H = {x ∈ F3

q : P (x) = 0}. Suppose that ∣H∣ ∼ q2 and H isnondegenerate(does not contain any planes). Then, we have the following extension estimateon H:

R∗(2→ 4) ≲ 1.

Here, we recall that ∣V ∣ ∼ q2 means that there exist C, c > 0 depending only on thedegree of the polynomial P (x) such that cq2 ≤ ∣V ∣ ≤ Cq2. We also notice that the norm ofthe extension operator depends only on the degree of V and on the ratio ∣V ∣/q2.

It seems that the conclusion of Theorem 1.8 implies the complete answer to extensionproblems associated with any nondegenerate homogeneous varieties in F3

q. At this moment,

we do not know whether the L2 − L4 estimate is improvable for some nondegenerate homo-geneous varieties in F3

q, although it is known from [14] that it gives the complete solutions to

the conical extension problem in F3q. However, we can prove that L2 − L4 estimate is sharp

if the variety H ⊂ F3q is some certain nondegenerate quadratic varieties.

Corollary 1.9. Let S be the surface defined as in (1.6). Suppose that d = 3 and there existsl ∈ Fq such that −aia−1

j = l2 for some i, j ∈ {1, 2, 3} with i ∕= j. Then R∗(p→ r) ≲ 1 if andonly if the exponents 1 ≤ p, r ≤ ∞ satisfy the following conditions: r ≥ 4 and r ≥ 2p/(p−1).

Since S ⊂ F3q always contains a line (one dimensional subspace), the statement of Corol-

lary 1.9 follows from the necessary condition (1.11) and the following lemma.

7

Lemma 1.10. For aj ∕= 0, j = 1, . . . , d, let S = {x ∈ Fdq : a1x21 + ⋅ ⋅ ⋅ + adx

2d = 0}. Suppose

that there exists l ∈ Fq such that −aia−1j = l2 for some i, j ∈ {1, . . . , d} with i ∕= j. If d ≥ 3

is odd, then R∗(p→ r) ≲ 1 only if r ≥ 2d−2d−2

.

The proof of Lemma 1.10 will be given in Appendix.

2. Discrete Fourier analysis

We denote by � the nontrivial additive character of Fq. For example, if q is prime, thenwe may take �(t) = e2�it/q where we identify t ∈ Fq with a usual integer. Recall that weendow the space Fdq with a normalized counting measure dx. Thus, given a complex valued

function f : Fdq → ℂ, the Fourier transform of f is defined by

f(m) =

∫Fdq�(−m ⋅ x)f(x) dx =

1

qd

∑x∈Fdq

�(−m ⋅ x)f(x),

where m is any element in the dual space of (Fdq , dx). Recall that the Fourier transform f

is actually defined on the dual space of (Fdq , dx). We shall endow the dual space of (Fdq , dx)

with a counting measure dm. We write (Fdq , dm) for the dual space of (Fdq , dx). Then, we also

see that the Fourier inversion theorem says that for every x ∈ (Fdq , dx),

f(x) =

∫Fdq�(x ⋅m)f(m) dm =

∑m∈Fdq

�(x ⋅m)f(m).

We also recall the Plancherel theorem: ∥f∥L2(Fdq ,dm) = ∥f∥L2(Fdq ,dx), which is same as∑m∈Fdq

∣f(m)∣2 =1

qd

∑x∈Fdq

∣f(x)∣2.

For instance, if f is a characteristic function on the subset E of Fdq , then the Planchereltheorem yields

(2.1)∑m∈Fdq

∣E(m)∣2 =∣E∣qd,

here, and throughout the paper, we identify the set E ⊂ Fdq with the characteristic functionon the set E.

3. Key lemmas and their proofs

In this section we establish the core lemmas which play an important role in provingour main results in this paper. First we review well-known properties of classical Gausssums in the finite field, and we apply them to derive the explicit form of the inverse Fouriertransform of the surface measure d� on the nondegenerate quadratic surface S defined as in(1.6). In the remainder of this paper, we fix an additive character � as the canonical additivecharacter of Fq and we always denote by � the quadratic character of Fq. For each t ∈ Fq,the Gauss sum Gt is defined by

Gt =∑

s∈Fq∖{0}

�(s)�(ts).

8

The absolute value of the Gauss sum is given by the relation

∣Gt∣ ={q

12 if t ∕= 0

0 if t = 0.

We also recall that

(3.1)∑s∈Fq

�(ts2) = �(t)G1 for any t ∕= 0.

For the nice proofs for the properties related to the Gauss sums, see Chapter 5 in [12]and Chapter 11 in [7]. If we complete the square and apply a change of variable, then thefollowing equation can be directly obtained from (3.1): for each a ∈ Fq ∖ {0}, b ∈ Fq

(3.2)∑s∈Fq

�(as2 + bs) = G1�(a)�

(b2

−4a

).

As a simple application of Gauss sum estimates, one can have the following explicit form of(d�)∨.

Lemma 3.1. Let d� be the surface measure on S defined as in (1.6). If d ≥ 3 is odd, thenwe have

(d�)∨(m) =

⎧⎨⎩qd−1∣S∣−1 if m = (0, . . . , 0)

0 if m ∕= (0, . . . , 0),m2

1

a1+ ⋅ ⋅ ⋅+ m2

d

ad= 0

Gd+11

q∣S∣ �(−a1 ⋅ ⋅ ⋅ ad)�(m2

1

a1+ ⋅ ⋅ ⋅+ m2

d

ad

)if

m21

a1+ ⋅ ⋅ ⋅+ m2

d

ad∕= 0.

If d ≥ 2 is even, then we have

(d�)∨(m) =

⎧⎨⎩qd−1∣S∣−1 +

Gd1∣S∣ (1− q

−1)�(a1 ⋅ ⋅ ⋅ ad) if m = (0, . . . , 0)Gd1∣S∣ (1− q

−1)�(a1 ⋅ ⋅ ⋅ ad) if m ∕= (0, . . . , 0),m2

1

a1+ ⋅ ⋅ ⋅+ m2

d

ad= 0

− Gd1q∣S∣�(a1 ⋅ ⋅ ⋅ ad) if

m21

a1+ ⋅ ⋅ ⋅+ m2

d

ad∕= 0,

Proof. Using the definition of the inverse Fourier transform and the orthogonality prop-erty of the nontrivial additive character � of Fq, we have

(d�)∨(m) = ∣S∣−1∑x∈S

�(x ⋅m)

= ∣S∣−1q−1∑x∈Fdq

∑s∈Fq

�(s(a1x

21 + ⋅ ⋅ ⋅+ adx

2d))�(x ⋅m)

= qd−1∣S∣−1�0(m) + ∣S∣−1q−1∑x∈Fdq

∑s ∕=0

�(s(a1x

21 + ⋅ ⋅ ⋅+ adx

2d))�(x ⋅m)

= qd−1∣S∣−1�0(m) + ∣S∣−1q−1∑s ∕=0

d∏j=1

∑xj∈Fq

�(sajx2j +mjxj).

Using the equality (3.2) yields that

(d�)∨(m) = qd−1∣S∣−1�0(m) +Gd1∣S∣−1q−1�(a1 ⋅ ⋅ ⋅ ad)

∑s ∕=0

�d(s)�

(− 1

4s

(m2

1

a1

+ ⋅ ⋅ ⋅+ m2d

ad

)).

9

Case I. Suppose that d ≥ 3 is odd. Then �d ≡ �, because � is the quadratic character.

Therefore, ifm2

1

a1+ ⋅ ⋅ ⋅+ m2

d

ad= 0, then the proof is complete, because

∑s∈Fq∖{0} �(s) = 0. On

the other hand, ifm2

1

a1+ ⋅ ⋅ ⋅ + m2

d

ad∕= 0, then the statement follows from using a change of

variable,− 14s

(m2

1

a1+ ⋅ ⋅ ⋅+ m2

d

ad

)→ s, and the facts that �(4) = 1, �(s) = �(s−1) for s ∕= 0, and

G1 =∑

s ∕=0 �(s)�(s).

Case II. Suppose that d ≥ 2 is even. Then �d ≡ 1. The proof is complete, because∑s ∕=0 �(as) = −1 for all a ∕= 0, and

∑s ∕=0 �(as) = (q − 1) if a = 0. □

The sharpness of the uniform size estimates of (d�)∨ can be given by Lemma 3.1.

Corollary 3.2. If d ≥ 3 is odd, then it follows that

(d�)∨(0, . . . , 0) = 1,

∣(d�)∨(m)∣ ≲ q−(d−1)

2 if m ∕= (0, . . . , 0).

On the other hand, if d ≥ 4 is even, then we have

(d�)∨(0, . . . , 0) = 1,

∣(d�)∨(m)∣ ≲ q−(d−2)

2 if m ∕= (0, . . . , 0).

Proof. Notice that the inverse Fourier transform of d� is given by

(d�)∨(m) =

∫S

�(x ⋅m)d� =1

∣S∣∑x∈S

�(x ⋅m)

where m ∈ (Fdq , dm). Therefore, it is clear that (d�)∨(0, . . . , 0) = 1 for all d ≥ 2. If we

compare this with the values (d�)∨(0, . . . , 0) by Lemma 3.1, then we see that ∣S∣ ∼ qd−1

for d ≥ 3. Since the absolute value of the Gauss sum G1 is exactly q1/2, the statements inCorollary 3.2 follow immediately from Lemma 3.1. □

The following dual extension estimate shall be used to prove the main averaging re-sult, Theorem 1.4. Hence, there is a connection between averaging problems and extensionproblems for nondegenerate quadratic surface S. It seems that there exists such connectionbetween these problems for nondegenerate homogeneous surfaces with any degree � ≥ 2.However, we shall not pursue this issue in this paper.

Lemma 3.3. For any subset E of (Fdq , dx) and bj ∕= 0 for j = 1, . . . , d, if d ≥ 4 is even, thenwe have ∑

m∈S

∣E(m)∣2 :=∑m∈S

∣∣∣∣∣q−d∑x∈E

�(−m ⋅ x)

∣∣∣∣∣2

≲ min{q−(d+1)∣E∣

d+2d , q−d∣E∣

},

where S = {m ∈ Fdq : b1m21 + ⋅ ⋅ ⋅+ bdm

2d = 0} ⊂ (Fdq , dm)

Proof. It is clear from the Plancherel theorem (2.1) that∑m∈S

∣E(m)∣2 ≤∑m∈Fdq

∣E(m)∣2 = q−d∣E∣.

It therefore remains to show that

(3.3)∑m∈S

∣E(m)∣2 :=∑m∈S

∣∣∣∣∣q−d∑x∈E

�(−m ⋅ x)

∣∣∣∣∣2

≲ q−(d+1)∣E∣d+2d .

10

Since the space (Fdq , dx) is isomorphic to the dual space (Fdq , dm) as an abstract group, we may

identify the space (Fdq , dx) with the dual space (Fdq , dm). Thus, they possess same algebraicstructures. In addition, we endow them with different measures: the counting measuredm for (Fdq , dm) and the normalized counting measure for (Fdq , dx). For these reasons, the

inequality (3.3) is essentially same as the following: for every subset E of (Fdq , dm)

(3.4)∑x∈S

q−2d∣E(x)∣2 ≲ q−(d+1)∣E∣d+2d ,

where S is considered as

S = {x ∈ Fdq : b1x21 + ⋅ ⋅ ⋅+ bdx

2d = 0} ⊂ (Fdq , dx) and E(x) =

∑m∈Fdq

�(−m ⋅ x)E(m).

By duality (1.9), the statement (1.13) in Theorem 1.6 implies that the following restrictionestimate holds: for every function g on (Fdq , dm),

∥g∥2L2(S,d�) ≲ ∥g∥2

L2dd+2 (Fdq ,dm)

.

If we take g(m) = E(m), then we have

1

∣S∣∑x∈S

∣E(x)∣2 ≲ ∣E∣d+2d

Since ∣S∣ ∼ qd−1, (3.4) follows and the proof of Lemma 3.3 is complete. □

We now derive a lemma related to a geometric property of nondegenerate homogeneousvariety H ⊂ F3

q. First, let us review the well-known Schwartz-Zippel lemma, which gives us

the information about the cardinality of any variety in Fdq . For a nice proof of the Schwartz-Zippel lemma below, see Theorem 6.13 in [12].

Lemma 3.4 (Schwartz-Zippel). Let P (x) ∈ Fq[x1, ⋅ ⋅ ⋅ , xd] be a nonzero polynomial of degree�. Then, we have

∣{x ∈ Fdq : P (x) = 0}∣ ≤ �qd−1.

Using the Schwartz-Zippel lemma, we obtain the following lemma which we shall needfor the proof of main extension result, Theorem 1.8.

Lemma 3.5. Let P (x) ∈ Fq[x1, x2, x3] be a nonzero homogeneous polynomial with degree �.Assume that H = {x ∈ F3

q : P (x) = 0} is a nondegenerate homogeneous variety with degree

�. Then, we have for every m ∈ F3q ∖ {(0, 0, 0)},∣H ∩ Πm∣ ≤ �q,

where Πm = {x ∈ F3q : m ⋅ x = 0} which is a hyperplane passing through the origin.

Proof. Since m ∕= (0, 0, 0), we may assume that m = (m1,m2,−1),

Πm = {x ∈ F3q : m1x1 +m2x2 − x3 = 0},

andH = {x ∈ F3

q : P (x) = 0}.Thus, we see that

H ∩ Πm = {(x1, x2,m1x1 +m2x2) ∈ F3q : P (x1, x2,m1x1 +m2x2) = 0}.

11

Put R(x1, x2) = P (x1, x2,m1x1 +m2x2). Then, it is clear that

∣H ∩ Πm∣ = ∣{(x1, x2) ∈ F2q : R(x1, x2) = 0}∣.

If R(x1, x2) is a nonzero polynomial, then the Schwartz-Zippel lemma tells us that ∣H∩Πm∣ ≤�q and we complete the proof. Now assume R(x1, x2) is a zero polynomial. Then, it followsthat R(x1, x2) = P (x1, x2,m1x1 +m2x2) = 0 for all x1, x2 ∈ Fq. This implies that the varietyH = {x ∈ F3

q : P (x) = 0} contains a plane m1x1 +m2x2 − x3 = 0, which contradicts to ourassumption that H is nondegenerate. Thus, the proof is complete. □

4. Proof of Theorem 1.4 (main result on averaging problems)

In this section, we restate Theorem 1.4 and prove it.

Theorem 1.4. Let S ⊂ Fdq be a nondegenerate quadratic surface defined as in (1.5) or (1.6).Suppose that d ≥ 4 is even. Then,

(4.1) A(p→ r) ≲ 1 if

(1

p,1

r

)∈ Ω ∖ {P1, P2},

where Ω denotes the convex hull of points (0, 0), (0, 1), (1, 1),

P1 =

(d2 − 2d+ 2

d(d− 1),

1

(d− 1)

)and P2 =

(d− 2

d− 1,d− 2

d(d− 1)

).

Moreover, if (1/p, 1/r) = P1 or P2, then

(4.2) A(p→ r) ≲ log q.

In particular, if S contains a d/2-dimensional subspace and (1/p, 1/r) /∈ Ω, then Lp − Lr

estimate is not possible.

Proof. First observe that the sharpness follows from the necessary condition (1.3).We now prove the statement (4.2). Applying the duality and the pigeonhole principle (seeLemma 6.1 in Appendix), it is enough to prove the following restricted strong-type estimate:

(4.3) ∥E ∗ d�∥Ld−1(Fdq ,dx) ≲ ∥E∥L

d(d−1)

d2−2d+2 (Fdq ,dx)

for all E ⊂ Fdq .

We now consider the Bochner-Riesz kernel K on (Fdq , dm) defined by K = (d�)∨− �0, where�0(m) = 1 if m = (0, . . . , 0), and 0 otherwise. Then, our task is to establish the followingtwo inequalities: for all E ⊂ Fdq ,

(4.4) ∥E ∗ �0∥Ld−1(Fdq ,dx) ≲ ∥E∥L

d(d−1)

d2−2d+2 (Fdq ,dx)

for all E ⊂ Fdq

and

(4.5) ∥E ∗ K∥Ld−1(Fdq ,dx) ≲ ∥E∥L

d(d−1)

d2−2d+2 (Fdq ,dx)

for all E ⊂ Fdq .

Since �0 = 1 and the total mass of Fdq is one, the inequality (4.4) follows immediately fromYoung’s inequality for convolution. In order to prove the inequality (4.5), we first derive thefollowing lemma.

12

Lemma 4.1. Let d� be the surface measure on the nondegenerate quadratic surface S ⊂(Fdq , dx) in (1.6). If the dimension d ≥ 4 is even, then for every set E ⊂ (Fdq , dx), we have

(4.6) ∥E ∗ K∥L∞(Fdq ,dx) ≲∣E∣qd−1

and

(4.7) ∥E ∗ K∥L2(Fdq ,dx) ≲

{q−d+ 1

2 ∣E∣ d+22d if 1 ≤ ∣E∣ ≤ q

d2

q−d+1∣E∣ 12 if qd2 ≤ ∣E∣ ≤ qd,

where K = (d�)∨ − �0.

Proof. The estimate (4.6) follows immediately from Young’s inequality and the obser-

vation that ∥K∥L∞(Fdq ,dx) ≲ q. Let us prove the estimate (4.7). Using Plancherel, we have

∥E ∗ K∥2L2(Fdq ,dx) = ∥EK∥2

L2(Fdq ,dm)

=∑m∈Fdq

∣E(m)∣2∣K(m)∣2 =∑

m∕=(0,...,0)

∣E(m)∣2∣(d�)∨(m)∣2,

where the last line follows from the definition of K and the fact that (d�)∨(0, . . . , 0) = 1.Since ∣S∣ ∼ qd−1, ∣�∣ ≡ 1, and the absolute value of the Gauss sum G1 is q1/2, using theexplicit formula for (d�)∨ in the second part of Lemma 3.1 shows that

∥E ∗ K∥2L2(Fdq ,dx)

∼ 1

qd−2

∑m ∕=(0,...,0):

m21

a1+⋅⋅⋅+m2

dad

=0

∣E(m)∣2 +1

qd

∑m ∕=(0,...,0):

m21

a1+⋅⋅⋅+m2

dad∕=0

∣E(m)∣2 = I + II.

From the Plancherel theorem (2.1), it is not difficult to see that

II ≤ 1

qd

∑m∈Fdq

∣E(m)∣2 = q−2d∣E∣.

Using the dual extension estimate (Lemma 3.3), the upper bound of I is obtained:

I ≲ min{q−2d+1∣E∣

d+2d , q−2d+2∣E∣

}.

Putting together above estimates yields

∥E ∗ K∥2L2(Fdq ,dx) ≲ min

{q−2d+1∣E∣

d+2d , q−2d+2∣E∣

}+ q−2d∣E∣

∼ min{q−2d+1∣E∣

d+2d , q−2d+2∣E∣

}.

By a direct calculation, we see that this estimate implies (4.7). The proof of Lemma 4.1 iscomplete. □

We now return to the proof of the inequality (4.5). Observe that the inequality (4.5)follows from interpolating the following inequalities: for all E ⊂ Fdq ,

(4.8) ∥E ∗ K∥L∞(Fdq ,dx) ≲ q∥E∥L1(Fdq ,dx)

13

and

(4.9) ∥E ∗ K∥L2(Fdq ,dx) ≲ q−d+3

2 ∥E∥L

2dd+2 (Fdq ,dx)

.

Since (4.8) is clear from (4.6), it remains to prove that (4.9) holds. However, the inequality(4.9) is also established by comparing the right-hand side of (4.7) and the right-hand side of(4.9). Thus, the statement (4.2) in Theorem 1.4 follows. In order to complete the proof ofTheorem 1.4, it remains to prove the statement (4.1). Notice that L∞−L∞, L∞−L1, and L1−L1 estimates are of strong-type estimates. In addition, recall from (1.8) in Corollary 1.3 thatLd/(d−1) − Ld is of a strong-type estimate, and observe that the point P3 := ((d− 1)/d, 1/d)is the midpoint of P1 and P2. By duality and the Riesz-Thorin interpolation theorem, ittherefore suffices to show that for every point (1/p, 1/r) between (1, 1) and P1, and betweenP1 and P3, we have

A(p→ r) ≲ 1.

We know from (4.3) that the point P1 yields a restricted strong-type estimate. Moreover, it isclear that the point (1, 1) also yields a restricted strong-type estimate. By usual interpolation,it therefore follows that for every point (1/p, 1/r) between (1, 1) and P1, the Lp−Lr estimateis a restricted strong-type inequality. This also holds for all points (1/p, 1/r) between P1

and P3. Therefore, the statement (4.1) follows from the Marcinkiewicz interpolation lemmabelow and we compete the proof of Theorem 1.4:

Lemma 4.2. Suppose 1 < pi < qi < ∞ for i = 0, 1, p0 < p1, q0 < q1, and T is of restrictedweak-type (pi, qi), i = 0, 1. 1 Then, T is of strong-type (p�, q�) for any 0 < � < 1, where1/p� = (1− �)/p0 + �/p1 and 1/q� = (1− �)/q0 + �/q1.

For a nice proof of Lemma 4.2, see [20].□

5. The proof of Theorem 1.8(main result on extension problems)

We shall apply the following well-known lemma to obtain L2 − L4 extension estimate,which is closely related to estimating the incidences between the variety and its nontrivialtranslations. For a complete proof of the following lemma, see both Lemma 5.1 in [14] andthe proof of Theorem 1.1 in [9].

Lemma 5.1. Let V be any algebraic variety in Fdq , d ≥ 2, with ∣V ∣ ∼ qd−1. Suppose that for

every � ∈ Fdq ∖ {(0, . . . , 0)}, ∑(x,y)∈V×V :x+y=�

1 ≲ qd−2.

Then, we have

R∗(2→ 4) ≲ 1.

Using Lemma 5.1, the following lemma shall give the complete proof of Theorem 1.8.

1We say that T is of restricted weak-type (p, r) if the following estimate holds:∣∣x ∈ Fdq : ∣TE(x)∣ > �}

∣∣ ≲ �−r∣E∣rp for all E ⊂ Fd

q , � > 0.

14

Lemma 5.2. Let P (x) ∈ Fq[x1, x2, x3] be a homogeneous polynomial. Suppose that thehomogeneous variety H = {x ∈ F3

q : P (x) = 0} does not contain any plane passing through

the origin. Then, we have that for every � ∈ F3q ∖ {(0, 0, 0)},

∣{(x, y) ∈ H ×H : x+ y = �}∣ ≲ q.

Proof. The first observation is that since H is a homogeneous variety, H is exactly theunion of lines passing through the origin. To see this, just note that if P (x) = 0 for somex ∕= (0, 0, 0), then P (tx) = 0 for all t ∈ Fq. Therefore, we can write

(5.1) H = ∪Nj=1Lj,

where N is a fixed positive integer, Lj denotes a line passing through the origin, and Li∩Lj ={(0, 0, 0)} for i ∕= j. From the Schwartz-Zippel lemma, it is clear that ∣H∣ ≲ q2. Thus, thenumber of lines, denoted by N , is ≲ q, because each line contains q elements. The secondimportant observation is that if H does not contain any plane passing through the origin,then for every m ∈ F3

q ∖ {(0, 0, 0)},(5.2) ∣H ∩ Πm∣ ≲ q,

where Πm = {x ∈ F3q : m ⋅ x = 0}. This observation follows from Lemma 3.5. We are ready

to prove our lemma. For each � ∕= (0, 0, 0), it suffices to prove that the number of commonsolutions of P (x) = 0 and P (�− x) = 0 is ≲ q. Since P (x) is a homogeneous polynomial, wesee that P (� − x) = 0 if and only if P (x− �) = 0. Therefore, we aim to show that for every� ∕= (0, 0, 0),

∣H ∩ (H + �)∣ ≲ q,

where H + � = {(x+ �) ∈ F3q : x ∈ H}. Now, fix � ∕= (0, 0, 0). From (5.1), we see that

∣H ∩ (H + �)∣ ≤N∑j=1

∣H ∩ (Lj + �)∣.

Notice that if � ∈ Lj, then Lj + � = Lj and so ∣H ∩ (Lj + �)∣ = q. However, there isat most one line Lj such that � ∈ Lj. Thus, it is enough to show that if � /∈ Lj, then∣H ∩ (Lj + �)∣ ≲ 1, because N ≲ q. However, this will be clear from (5.2). To see this,first notice that if (0, 0, 0) ∕= � /∈ Lj, then the line Lj + � does not pass through the origin,because the line Lj passes through the origin. Thus, the line Lj + � is different from all linesLk in H = ∪Nk=1Lk, and so there is at most one intersection point of the line Lj + � with eachline in H. Next, consider the unique plane Πm which contains the line Lj + �. Then, (5.2)implies that at most few lines in H lie in the plane Πm containing the line Lj + �. Thus, weconclude that ∣H ∩ (Lj + �)∣ ≲ 1 for � /∈ Lj, and the proof is complete. □

6. Appendix

6.1. Proof of Lemma 1.10. We prove Lemma 1.10. Without loss of generality we mayassume that −ad−1a

−1d = l2 for some l ∈ Fq. Then the set S is given by

S = {x ∈ Fdq : a1x21 + ⋅ ⋅ ⋅+ ad−2x

2d−2 − ad(lxd−1 + xd)(lxd−1 − xd) = 0}.

Since the mapping property of extension operators associated with S is invariant under thenon-singular linear transform of the surface S, we may assume

S = {x ∈ Fdq : a1x21 + ⋅ ⋅ ⋅+ ad−2x

2d−2 − xd−1xd = 0}.

15

Let D = {s ∈ Fq ∖ {0} : s is a square number}. Then it is clear that ∣D∣ = (q − 1)/2 ∼ q.Now, define a set

Ω =

{m ∈ Fd−1

q × D : md−1 =a−1

1 m21 + ⋅ ⋅ ⋅+ a−1

d−2m2d−2

4md

}.

Observe that ∣Ω∣ = qd−2∣D∣ ∼ qd−1. We test (1.9) with the characteristic function on the setΩ ⊂ (Fdq , dm). We have

(6.1) ∥Ω∥Lr′ (Fdq ,dm) = ∣Ω∣1r′ ∼ q

d−1r′ .

Let us estimate the quantity ∥Ω∥Lp′ (S,d�). For each x ∈ S with xd−1 ∕= 0, we have

Ω(x) =

∫Fdq

Ω(m)�(−m ⋅ x)dm =∑m∈Ω

�(−m ⋅ x)

=∑

m1,...,md−2∈Fq

∑md∈D

�(ℙx1,...,xd(m1, . . . ,md−2,md)),

where ℙx1,...,xd(m1, . . . ,md−2,md) = −m1x1 − ⋅ ⋅ ⋅ −md−2xd−2 −(a−11 m2

1+⋅⋅⋅+a−1d−2m

2d−2

4md

)⋅ xd−1 −

mdxd. Therefore, the completing square method (3.2) yields that for each x ∈ S with xd−1 ∕=0,

Ω(x) = Gd−21 �(a−1

1 ⋅⋅ ⋅ ⋅⋅a−1d−2)�d−2

(−xd−1

4

) ∑md∈D

�d−2(m−1d )�

(md

(a1x

21 + ⋅ ⋅ ⋅+ ad−2x

2d−2

xd−1

− xd))

.

Since �d−2(m−1d ) = 1 for md ∈ D, and

a1x21+⋅⋅⋅+ad−2x2d−2

xd−1− xd = 0 for x ∈ S with xd−1 ∕= 0 we

see that for x ∈ S with xd−1 ∕= 0,

∣Ω(x)∣ = ∣Gd−21 ∣∣D∣ ∼ q

d2 .

Using this estimate we have

(6.2) qd2 ∼

⎛⎝ 1

∣S∣∑

x∈S:xd−1 ∕=0

qdp′2

⎞⎠ 1p′

≲ ∥Ω∥Lp′ (S,d�).

From (6.1) and (6.2), the proof of Lemma 1.10 is complete.

6.2. The pigeonhole principle.

Lemma 6.1. Suppose that for every subset E of Fdq it satisfies that

(6.3) ∥E ∗ d�∥Lr(Fdq ,dx) ≲ ∥E∥Lp(Fdq ,dx).

Then for every function f defined on (Fdq , dx), we have

∥f ∗ d�∥Lr(Fdq ,dx) ≲ log q ∥f∥Lp(Fdq ,dx).

Proof. Without loss of generality, we may assume that f ≥ 0 and ∥f∥∞ = 1. For eachnonnegative integer k, define Ek = {x ∈ Fdq : 2−k−1 < f(x) ≤ 2−k}. Then the function f canbe decomposed as f =

∑∞k=0 fk where fk = f ⋅Ek. From the definition of fk, we see that for

every nonnegative integer k,

(6.4) ∥f∥Lp(Fdq ,dx) ≥ ∥fk∥Lp(Fdq ,dx) ≥ 2−k−1∥Ek∥Lp(Fdq ,dx)

16

Since ∥f∥∞ = 1, we also have

(6.5) ∥f∥Lp(Fdq ,dx) ≥ q−dp ≥ 2−(N+1),

where N is the nonnegative integer satisfying

d log q

p log 2− 1 ≤ N <

d log q

p log 2.

We now estimate the quantity ∥f ∗ d�∥Lr(Fdq ,dx). It follows that

∥f ∗ d�∥Lr(Fdq ,dx) ≤ ∥N∑k=0

fk ∗ d�∥Lr(Fdq ,dx) + ∥∞∑

k=N+1

fk ∗ d�∥Lr(Fdq ,dx)

≤ N max0≤k≤N

∥fk ∗ d�∥Lr(Fdq ,dx) + 2−(N+1)∥1 ∗ d�∥Lr(Fdq ,dx).

From the definition of fk and the observation that ∥1 ∗ d�∥Lr(Fdq ,dx) = 1, we see that aboveexpression is dominated by the quantity

N max0≤k≤N

2−k∥Ek ∗ d�∥Lr(Fdq ,dx) + 2−(N+1).

From the hypothesis (6.3) and the inequalities (6.4), (6.5), we therefore obtain that

∥f ∗ d�∥Lr(Frq ,dx) ≲ N max0≤k≤N

2−k∥Ek∥Lp(Fdq ,dx) + 2−(N+1)

≤ (2N + 1)∥f∥Lp(Fdq ,dx).

Since 2N + 1 ∼ log q, we complete the proof of Lemma 6.1. □

References

[1] A. Carbery, Restriction implies Bochner-Riesz for paraboloids, Math. Proc. Cambridge Philos. Soc. 111(1992), no. 3, 525–529. 5

[2] T. Cochrane, Exponential sums and the distribution of solutions of congruences, Inst. of Math., AcademiaSinica, Taipei, (1994). 3

[3] A. Carbery, B. Stones, and J. Wright, Averages in vector spaces over finite fields, Math. Proc. Camb.Phil. Soc. (2008), 144, 13, 13–27. 1, 2

[4] L. DeCarli and A. Iosevich, A restriction theorem for flat manifolds of codimension two, Illinois Journalof Mathematics, Volume 39 (1995), Number 4, 576–585. 7

[5] A. Iosevich and D. Koh, Extension theorems for paraboloids in the finite field setting, Math. Z. 266 (2010),no.2, 471–487. 4

[6] A. Iosevich and D. Koh, Extension theorems for spheres in the finite field setting, Forum Math. 22 (2010),no.3, 457–483. 4

[7] H. Iwaniec and E. Kowalski, Analytic Number Theory, Colloquium Publications, 53 (2004). 9[8] A. Iosevich and E. Sawyer, Sharp Lp−Lq estimates for a class of averaging operators, Ann. Inst. Fourier,

Grenoble, 46, 5 (1996), 1359–1384. 2[9] D. Koh and C. Shen, Sharp extension theorems and Falconer distance problems for algebraic curves in

two dimensional vector spaces over finite fields , Revista Matematica Iberoamericana (to appear), (2011).14

[10] W. Littman, Lp − Lq estimates for singular integral operators, Proc. Symp. Pure Math., 23 (1973),479–481. 2

[11] A. Lewko and M. Lewko, Restriction estimates for the paraboloid over finite fields, preprint(www.arxiv.org), (2010). 4

[12] R. Lidl and H. Niederreiter, Finite fields, Cambridge University Press, (1997). 3, 9, 11[13] G. Mockenhaupt, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 25(1991),

31–36. 517

[14] G. Mockenhaupt, and T. Tao, Restriction and Kakeya phenomena for finite fields, Duke Math. J.121(2004), no. 1, 35–74. 1, 4, 5, 6, 7, 14

[15] R. Strichartz, Convolutions with kernels having singularities on the sphere, Trans. Amer. Math. Soc.,148 (1970), 461–471. 2

[16] E. M. Stein, Lp boundedness of certain convolution operators, Bull. Amer. Math. Soc., 77 (1971), 404–405. 2

[17] E. M. Stein, Harmonic Analysis, Princeton University Press (1993). 7[18] T. Tao, A sharp bilinear restriction estimate for paraboloids, Geom. Funct. Anal. 13 (2003), 1359–1384.

5[19] T. Tao, Recent progress on the restriction conjecture, Fourier analysis and convexity, 217–243,

Appl. Number. Harmon. Anal., Birkhuser Boston, Boston, MA 2004. 5[20] T. Tao, Tao’s lecture notes for Week 4, (http://www.math.ucla.edu/ tao/254a.1.01w/). 14[21] T. Wolff, A sharp bilinear cone restriction estimate, Annals of Math. 153 (2001), 661–698. 5

Department of Mathematics, Michigan State University, East Lansing, MI 48824 USAE-mail address: koh@math.msu.edu

Department of Mathematics and Statistics, McMaster University, Hamilton, CanadaE-mail address: shenc@umail.iu.edu

18