EXTENSION THEOREMS FOR THE FOURIER TRANSFORM ASSOCIATED WITH NON-DEGENERATE QUADRATIC SURFACES IN VECTOR SPACES OVER FINITE FIELDS

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1 EXTENSION THEOREMS FOR THE FOURIER TRANSFORM ASSOCIATED WITH NON-DEGENERATE QUADRATIC SURFACES IN VECTOR SPACES OVER FINITE FIELDS ALEX IOSEVICH AND DOOWON KOH Abstract. We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas-Stein exponent. We also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimates for Gauss and Kloosterman sums and their variants play an important role. 1. Introduction Let S be a subset of R d and dσ a positive measure supported on S. Then one may ask that for which values of p and r does the estimates (1.1) fdσ L r (R d ) C p,r f L p (S,dσ) for all f L p (S, dσ) hold? This problem is known as the extension theorems. See, for example, [5],[19],[15],[6],[17], and the references contained therein on recent progress related to this problem and its analogs. In the case of p = in (1.1), Strichartz [16] gave a complete solution when S is a quadratic surface given by S = {x R d : Q(x) = j}, where Q(x) is a polynomial of degree of two with real coefficients and j is a real constant. In this paper, we study the analogous extension operaters given by quadratic forms in the finite field setting, building upon earlier work of Mockenhaupt and Tao [13] for the paraboloid in vector spaces over finite fields. We begin with some notation and definitions to describe our main results. Let F q be a finite field of characteristic char(f q ) > with q elements, and let F d q be a d-dimensional vector space over F q. Given a Date: 1991 Mathematics Subject Classification.?,? Partially supported by grant? 1

2 ALEX IOSEVICH AND DOOWON KOH function f : F d q C, d 1, define the Fourier transform of f by the formula f(m) = q d x F d q χ( x m)f(x) where χ is a non-trivial additive character on F q. When F q = Z/qZ for some prime q, we could take χ(t) = e πit/q, and the calculations in the paper are independent of the exact choice of the character. Recall that the Fourier inversion theorem is given by f(x) = m F d q χ(x m) f(m). Also recall that the Plancherel theorem says in this context that f(m) = q d f(x). m F d q x F d q Let S F d q be an algebraic variety in F d q. We denote by dσ normalized surface measure on S defined by the relation fdσ(m) = 1 χ( x m)f(x), #S x S where #S denotes the number of elements in S. In other words, q d σ(x) = (#S) 1 S(x). Here, and throughout the paper, E(x) denotes the characteristic function, χ E, of the subset E of F d q. We therefore denote by Edσ the measure χ E dσ. For 1 p, r <, define and f p L p (F d q,dx) = q d x F d q f r L r (F d q,dm) = m F d q f p L p (S,dσ) = 1 #S f(x) p, f(m) r f(x) p. Similarly, denote by f L the maximum value of f. Observe that the measure on the space variables, dx, is the normalized measure obtained by dividing the counting measure by q d, whereas the measure on the phase variables, dm, is just the usual counting measure. These normalizations are chosen in such a way that the Plancherel inequality takes the familiar form f L (F d q,dm) = f L (F d q,dx). x S

3 EXTENSION THEOREMS 3 We now define the non-degenerate quadratic surfaces in F d q in the usual way. Let x = (x 1, x,, x d ) F d q. Denote by Q(x) a homogeneous polynomial in F q [x 1,, x d ] of degree. Since char(f q ) > throughout this paper, we can express Q(x) in the form Q(x 1, x,, x d ) = d a ij x i x j with a ij = a ji. i,j=1 If the d d matrix {a ij } is invertible, we say that the Polynomial Q(x) is a non-degenerate quadratic form over F q. For each j F q = F q \ {0}, the multiplicative group of F q, consider a set S j in F d q given by (1.) S j = {x F d q : Q(x 1,, x d ) = j}, where Q(x) is a non-degenerate quadratic form. We call such a set S j a non-degenerate quadratic surface in F d q. For example, the sphere S d 1 = {x F d q : x 1 + x + + x d = 1} is a non-degenerate quadratic surface in F d q Extension of the Fourier transform. Let 1 p, r. We define R (p r) to be the best constant such that the extension estimate fdσ L r (F d q,dm) R (p r) f L p (S j,dσ) holds for all functions f on S j. The main goal of this paper is to determine the set of exponents p and r such that R (p r) C p,r <, where C p,r is independent of the size of F q. We note that (1.3) R (p 1 r) R (p r) for p 1 p, and R (p r 1 ) R (p r ) for r 1 r, which will allow us to reduce the analysis below to certain endpoint estimates. Let S be an algebraic variety in F d q with #S q k for some 0 < k < d. Here, and throughout the paper, X Y means that there exists C > 0, independent of q such that X CY, and X Y means both X Y and Y X. Mockenhaupt and Tao [13] proved that R (p r) is uniformly bounded (O(1) with constants independent of the size of F q ) only if (1.4) r d k and r dp k(p 1). For the detailed proofs of these assertions, see ([13], page 41-4).

4 4 ALEX IOSEVICH AND DOOWON KOH Mockenhaupt and Tao also showed that R ( r) is uniformly bounded whenever (1.5) r d + d 1 if S = {(x, x x) : x F d 1 q }, an analog of the Euclidean paraboloid. Moreover, when d = 3 and 1 is not a square in F q, they improved the result in (1.5) by showing that for each ε > 0 there exists C ɛ > 0 such that ( ) ( 8 (1.6) R and R 18 5 ) C ɛ q ɛ,. If we replaced the paraboloid by a general non-degenerate quadratic surface, the extension problem becomes more complicated, in part because the Fourier transform of quadratic surfaces cannot be computed by simply considering the Gauss sums, as was pointed out by the authors in [13]. Using generalized Kloosterman sums, we estimate the decay of the Fourier transform of non-degenerate quadratic surfaces. As a result, we obtain Theorem 1 below which gives the same exponents as (1.5) (see the FIGURE 1). Theorem 1. Let S j be a non-degenerate quadratic surface in F d q defined as in (1.). If d and r d+ d 1, then R ( r) 1. In the case d =, Mochenhaupt and Tao [13] showed that the necessary conditions for the boundedness of R (p r) in (1.4) are also sufficient when S is the parabola. Theorem below implies that this also holds in the case when S is a non-degenerate quadratic curve. To see this, observe from Corollary 10 that #S q for d =. Thus the necessary conditions in (1.4) take the form (1.7) r 4 and r p p 1. Combining (1.3) with Theorem below, we see that (1.8) R (p 4) 1 for p. By direct estimation, we have (1.9) R (p ) 1 for 1 p. Interpolating (1.8) and (1.9), we see that the necessary conditions given by (1.7) are in fact sufficient as we claim once we establish the following result. Theorem. Let d. Let S j be the non-degenerate quadratic surface in F d q defined as in (1.). Then we have R ( 4) 1.

5 EXTENSION THEOREMS 5 Observe that Theorem is stronger in two dimensions. Theorem 1 and Theorem are the same in three dimensions, and Theorem 1 is stronger in dimension four and higher. Using incidence theory, we are able to improve the exponents above in a restricted setting. See Theorem 3 and FIGURE 1 for extension theorems restricted to big sets, and see also Theorem 4, FIGURE, 3, and 4 for extension theorems restricted to small sets. These results are analogous to those obtained by Mockenhaupt and Tao as described in (1.6) above. While the aforementioned authors use combinatorial methods to prove their incidence theorems, we use Fourier analytic methods which eventually reduce proofs to the estimates for Kloosterman and related sums. Theorem 3. Let S j be a non-degenerate quadratic surface in F d q and E be a subset of S j. Then we have the following estimate (1.10) Êdσ L 4 (F d q,dm) E L 4 3 (Sj,dσ) for q d+1 #E q d 1. Figure 1. Tomas-Stein exponent and extension estimates in a restricted setting to big sets (q d+1 #E q d 1 ) Theorem 4. Let S j be a non-degenerate quadratic surface in F d q and E be a subset of S j. Then for every p 0, we have the following estimates (1.11) Êdσ L r (F d q,dm) E L p (S j,dσ) for 1 #E q d 1 where the exponents p and r are given by and p (6d )p 0 8d + 8 (3d 5)p 0 4d + 1 and r (6d )p 0 8d + 8 (3d 3)p 0 4d + 4, (1.1) Êdσ L r (F d q,dm) E L p (S j,dσ) for 1 #E q d+1

6 6 ALEX IOSEVICH AND DOOWON KOH where the exponents p and r are given by p (6d 10)p 0 8d + 4 (3d 9)p 0 4d + 0 and r (6d 10)p 0 8d + 4 (3d 7)p 0 4d + 1. Figure. Extension estimates in a restricted setting (1 #E q d 1 ) Figure 3. Extension estimates in a restricted setting (1 #E q d+1, d > 5). Classical bounds on exponential sums and consequences In this section, we shall estimate the decay of Fourier transform of the characteristic functions on non-degenerate quadratic surfaces in F d q using the classical bounds on exponential sums. To do this, we first introduce the well known theorems for exponential sums. The following theorem is a well known estimate for Gauss sums.

7 EXTENSION THEOREMS 7 Figure 4. Extension estimates in a restricted setting (1 #E q d+1, d < 5) Theorem 5. Let χ be a non-trivial additive character of F q, and ψ a multiplicative character of F q. It follows that G a (χ, ψ) = χ(at)ψ(t) = O(q 1 ), a F q. t >F q Proof. If ψ = 1, the result is obvious, so we may assume that ψ is a non-trivial multiplicative character of F q. We Have G a (χ, ψ) = χ(at as)ψ(ts 1 ) t F q s F q Thus = t F q = t F q ψ(t) s F q χ(ast as) ( ψ(t) 1 + χ(st s) s F q = ψ(t) + χ((t 1)s) t F q t F q s F q = 0 + q = q. ) and the proof is complete. G a (χ, ψ) = q 1 The following theorem gives us the relation between more general exponential sums and Gauss sums in Theorem 5. For a nice proof, see [1].

8 8 ALEX IOSEVICH AND DOOWON KOH Theorem 6. Let χ be a non-trivial additive character of F q, n N, and ψ a multiplicative character of F q of order h = gcd(n, q 1). Then h 1 χ(ts n ) = ψ k (t)g(ψ k, χ) s F q k=1 for any t F q, where G(ψ k, χ) = ψ k (s)χ(s). s F q The following theorem is well known as the estimation of the Salié sum, often referred to as the twisted Kloosterman sum. See [14]. Theorem 7. Let ψ be a multiplicative character of order two of F q, q odd, and a, b F q. Then for any additive character χ of F q, ψ(t)χ(at + bt 1 ) q 1. t F q The following is a classical estimate for Kloosterman sums due to Wey [18]. See also [1]. Theorem 8. If χ is a non-trivial additive character of F q and a, b F q are not both 0, then we have χ(at + bt 1 ) q 1. t F q With the same notation as above, we have the following estimate on the Fourier transform of the characteristic function of a non-degenerate quadratic surface. Lemma 9. Let F q, q odd, be a finite field. Then Ŝj(m) = q d χ( x m) q d+1 x S j if m (0,, 0), and Ŝ j (0,, 0) q 1. From Lemma 9, we obtain the following corollary. Corollary 10. #S j q d 1. Proof. Using the second part of Lemma 9, we have and the result follows. Ŝ j (0,, 0) = q d x F d q S j (x) q 1,

9 EXTENSION THEOREMS 9.1. Proof of Lemma 9. We first observe that Ŝ j (m) = q d x S j χ( x m) = q d x F d q χ( x m)q 1 t F q χ(t(q(x) j)) = q 1 δ 0 (m) + q d 1 t F q χ( jt) x F d q χ(tq(x) x m) where δ 0 (m) = 1 if m = (0,, 0) and δ 0 (m) = 0 otherwise. To complete the proof of Lemma 9, it suffices to show that for j 0, m F d q, (.1) D(j, m) q d+1 where (.) D(j, m) = t F q χ( jt) x F d q Let W t (m) = x F d q χ(tq(x) x m). χ(tq(x) x m) for m F d q, t F q. We shall need the following theorem (see [1]). Theorem 11. Every quadratic form Q(x) = d i,k=1 a ik x i x k over F q, q odd, can be transformed into a diagonal form a 1 x a d x d over F q by means of a nonsingular linear substitution of indeterminates. Moreover if Q(x) is a non-degenerate quadratic form, then a i 0 for all i = 1,,, d. Using Theorem 11, we may write that for some m = (m 1,, m d ) Fd q, and a i F q for all i = 1,,, d, W t (m) = x F d q χ(t x a + x m ), where m F d q is determined by m F d q and x a is given by x a = a 1 x a d x d.

10 10 ALEX IOSEVICH AND DOOWON KOH Since χ is an additive character of F q, we have d W t (m) = χ(ta k x k + m kx k ) x k F q = = k=1 d k=1 d k=1 Using Theorem 6, we see that x k F q χ(ta k (x k + (ta k ) 1 m k) (4ta k ) 1 m k χ( (4ta k ) 1 m k ) χ(ta k x k). x k F q x k F q χ(ta k x k) = ψ 1 (ta k )G(χ, ψ) where ψ is a multiplicative character of F q of order two and G(χ, ψ) = χ(s)ψ(s). Thus we obtain that s F q (.3) W t (m) = ψ d (t)ψ 1 (a 1 a d )(G(χ, ψ)) d ) d χ( (4ta k ) 1 m k ) k=1 d = ψ d (t)ψ 1 (a 1 a d )(G(χ, ψ)) d χ(t 1 (4a k ) 1 m k ). k=1 Combining above fact in (.3) with (.), we obtain that (.4) D(j, m) = ψ 1 (a 1 a d )(G(χ, ψ)) d t F q where M is given by M = d (4a k ) 1 m k. k=1 χ( jt + t 1 M)ψ d (t), Since ψ is a multiplicative character of order two, we see that ψ d = 1 for d even, and ψ d = ψ for d odd. Therefore, in order to get the inequality in (.1), we can apply Theorem 5 and 7 to (.4) for d odd. On the other hand, if d is even, we can apply Theorem 5 and 8 to (.4) because j 0. This completes the proof of Lemma Proof of the Tomas-Stein exponent (Theorem 1) Theorem 1 is a result from Lemma 9 and Lemma 6.1 in [13]. We first introduce Lemma 6.1 in [13]. Let S be an algebraic variety in F d q with a normalized surface measure dσ. We introduce the Bochner-Riesz kernel K(m) := dσ(m) δ 0 (m)

11 EXTENSION THEOREMS 11 where δ 0 (m) = 1 if m = (0,, 0) and δ 0 (m) = 0 otherwise. We need the following theorem. For a nice proof, see Lemma 6.1 in [13]. Theorem 1. Let p, r, and F d q be a d-dimensional vector space over F q. Suppose that K L (F d q,dm) = dσ δ 0 L (F d q,dm) q d for some 0 < d < d. Then ( for any 0 < θ < 1, we have R p r ) 1 + R (p r) θ q d(1 θ) 4. θ We are working with a non-degenerate quadratic surface S j in F d q. We now check that (3.1) K L (F d q,dm) q (d 1). In fact, if m = (m 1, m,, m d ) (0,, 0) then we have K(m) = dσ(m) = (#S j ) 1 x S j χ( x m) = (#S j ) 1 x F d q = (#S j ) 1 q d Ŝ j (m). From Corollary 10 and Lemma 9, we have χ( x m)s j (x) (#S j ) q d 1 and Ŝj(m) q d+1 for m (0,, 0). We therefore obtain that On the other hand, we have K(m) q d 1 for m (0,, 0). K(0,, 0) = dσ(0,, 0) 1 = 0. Thus the inequality in (3.1) holds. We now claim that (3.) R ( ) q 1. To justify above claim, we shall show that (3.3) fdσ L (F d q,dm) q 1 f L (S j,dσ) for all functions f on S j. We first note that fdσ(m) = (#S j ) x S j χ( x m)f(x) y S j χ(y m)f(y) = (#S j ) x,y S j χ((y x) m)f(x)f(y).

12 1 ALEX IOSEVICH AND DOOWON KOH We have fdσ ( L (F d q,dm) = fdσ(m) ) 1 m F d q = (#S j ) 1( ) 1 χ((y x) m)f(x)f(y) = (#S j ) 1 q d x,y S j m F d q ( x S j f(x) ) 1 = (#S j ) 1 q d (#Sj ) 1 f L (S j,dσ) q 1 f L (S j,dσ) In the last equality, we used the fact that #S j q d 1. Thus our claim in (3.) is proved. Using Theorem 1 with (3.1) and (3.), we obtain that for any 0 < θ < 1, R ( θ ) 1 + R ( ) θ q (d 1)(1 θ) q θ q (d 1)(1 θ) 4. Taking 0 < θ d 1 d+1, we have R ( ) 1. θ Thus Theorem 1 is proved with r = θ. 4. Proof of the L L 4 estimate (Theorem ) To prove Theorem, we make the following reduction. Lemma 13. Let S j be a non-degenerate quadratic surface in F d q defined as in (1.). Suppose that for any x (F d q) = F d q \ (0,, 0), we have 1 q d. Then for d, Proof. We have to show that {(α,β) S j S j:α+β=x} R ( 4) 1. fdσ L 4 (F d q,dm) f L (S j,dσ) for all functions f on S j. Using Plancherel, we have fdσ L 4 (F d q,dm) = fdσ fdσ 1 L (F d q,dm) = fdσ fdσ 1 L (F d q,dx)

13 EXTENSION THEOREMS 13 and so it suffices to show that fdσ fdσ L (F d q,dx) f 4 L (S. j,dσ) It follows that fdσ fdσ L (F d q,dx) = q d fdσ fdσ(0,, 0) + fdσ fdσ L ((F d q ),dx) Thus it will suffice to show that (4.1) q d fdσ fdσ(0,, 0) f 4 L (S j,dσ) and (4.) fdσ fdσ L ((F d q ),dx) f 4 L (S j,dσ) We first show that the inequality in (4.1) holds. We have fdσ fdσ(0,, 0) m F d q fdσ(m) = (#S j ) q d x S j f(x) = (#S j ) 1 q d f L (S j,dσ) q f L (S j,dσ). Thus the inequality in (4.1) holds because d. It remains to show that the inequality in (4.) holds. Without loss of generality, we may assume that f is positive. Using the Cauchy Schwartz inequality, we see that (4.3) fdσ fdσ(x) = (#S j ) q d (#S j ) q d( {(α,β) S j S j:α+β=x} {(α,β) S j S j:α+β=x} = (dσ dσ) 1 (x)(f dσ f dσ) 1 (x). f(α)f(β) ) 1 ( 1 {(α,β) S j S j:α+β=x} ) 1 f (α)f (β) From our hypothesis and the fact that #S j q d 1, we obtain that for x (0,, 0), (4.4) dσ dσ(x) q d From Fubini s theorem, we also have {(α,β) S j S j:α+β=x} (4.5) f dσ f dσ L 1 (F d q,dx) = f 4 L (S j,dσ).

14 14 ALEX IOSEVICH AND DOOWON KOH Using Hölder inequality and estimates (4.3), (4.4), and (4.5), we obtain that fdσ fdσ L ((F d q ),dx) = (fdσ fdσ) L 1 ((F d q ),dx) (dσ dσ) (f dσ f dσ) L 1 ((F d q ),dx) dσ dσ L ((F d q ),dx) f dσ f dσ L 1 ((F d q ),dx) f 4 L (S j,dσ). Thus the inequality in (4.) holds and so the proof of Lemma 13 is complete. We now prove Theorem. By Lemma 13, it is enough to show that for any x (F d q), d, (4.6) 1 q d {(α,β) S j S j:α+β=x} where S j is the non-degenerate quadratic surface in F d q. Using Theorem 11, we may assume that the non-degenerate quadratic surface in F d q is given by S j = {y F d q : a 1 y a d y d = j 0} for all a k 0, k = 1,,, d. Therefore the left hand side of the equation in (4.6) can be estimated by the number of common solutions α = (α 1,, α d ) in F d q of the equations (4.7) a 1 α1 + + a d αd = j d a 1 x 1 α a d x d α d = a k x k for x = (x 1,, x d ) (0,, 0) and a k 0 for all k = 1,,, d. Note that a k x k 0 for some k = 1,,, d because x (0,, 0) and a k 0. Thus a routine algebraic computation shows that the number of common solutions of equations in (4.7) is less than equal to q d. This means that the inequality in (4.6) holds and so we complete the proof of Theorem. 5. Incidence theorems and the proof of Theorem 3 and Theorem 4 The purpose of this section is to develop the incidence theory needed to prove both Theorem 3 and Theorem 4. Theorem 14. Let S j be a non-degenerate quadratic surface in F d q defined as before. If E is any subset of S j, then we have 1 (#E) q 1 + (#E)q d 1 {(x,y) E E:x y+z S j} k=1 for all z F d q where the bound is independent of z F d q.

15 EXTENSION THEOREMS 15 Proof. Fix E S j. For each z F d q, consider 1 {(x,y) E E:x y+z S j} = E(x)E(y)S j (x y + z) where and = (x,y) F d q Fd q (x,y) F d q Fd q E(x)E(y) m F d q = q d m F d q II = q d χ(m (x y + z))ŝj(m) Ê(m) χ(m z)ŝj(m) = I + II I = q d Ê(0,, 0) Ŝ j (0,, 0) m (0,,0) Ê(m) χ(m z)ŝj(m). Using Lemma 9 and Plancherel, we obtain that and II q d q d+1 q d q d+1 This completes the proof. I (#E) q 1, m (0,,0) q d x F d q Ê(m) E(x) = q d 1 (#E). Corollary 15. Let S j be a non-degenerate quadratic surface in F d q and E be any subset of S j. Then we have 1 min{(#e) 3, (#E) 3 q 1 + (#E) q d 1 }. {(x,y,z,s) E 4 :x+z=y+s} Proof. Since E is a subset of S j, we have {(x,y,z,s) E 4 :x+z=y+s} 1 z E {(x,y) E :x y+z S j} Thus Corollary 15 is the immediate result from Lemma 14 and the obvious fact that 1 (#E) 3. {(x,y,z,s) E 4 :x+z=y+s} 1.

16 16 ALEX IOSEVICH AND DOOWON KOH We are now ready to prove Theorem 3. We first expand the left-hand side of the inequality in (1.10). It follows that ( Êdσ 1 L 4 (F d q,dm) = Êdσ(m) 4) 4 (5.1) m F d q ( 1 = χ( x m)e(x) 4 ) 1 4 #S j x S j m F d q = 1 #S j ( x,y,z,s E S j m F d q = q d 4 #S j ( {(x,y,z,s) E 4 :x+z=y+s} ) 1 4 χ((x y + z s) m) ) Since q d+1 #E q d 1 from the hypothesis, we use Corollary 15 to obtain that (5.) 1 (#E) 3 q 1. {(x,y,z,s) E 4 :x+z=y+s} Combining (5.1) and (5.), we have (5.3) Êdσ L 4 (F d q,dm) q d 1 4 (#E) 3 4 #S j. On the other hand, by expanding the right-hand side of the inequality in (1.10), we see that ( #E ) 3 (5.4) E 4 = 4 L 3. (Sj,dσ) #S j Since #S j q d 1 by Corollary 10, comparing (5.3) with (5.4) yields the inequality in (1.10) and completes the proof. Lemma 16. Let S j be a non-degenerate quadratic surface in F d q and E be a subset of S j. For p 0,we have the following estimates (5.5) Êdσ L 4 (F d q,dm) q 3d d 1 p 0 E L p 0 (Sj,dσ) for 1 #E q d 1 (5.6) and Êdσ L 4 (F d q,dm) q 3d d 3 p 0 E L p 0 (Sj,dσ) for q d 1 #E q d+1, (5.7) Êdσ L 4 (F d q,dm) q 3d d 3 p 0 E L p 0 (Sj,dσ) for 1 #E q d+1.

17 EXTENSION THEOREMS 17 Proof. We first prove the inequality in (5.5). Since 1 #E q d 1, using Corollary 15 we have 1 (#E) 3. {(x,y,z,s) E 4 :x+z=y+s} Combining this with the fact in (5.1), we obtain that (5.8) Êdσ L 4 (F d q,dm ) q d 4 (#E) 3 4 #S j. As before, we note that (5.9) E L p 0 (Sj,dσ) ( #E #S j ) 1 p 0. From (5.8) and (5.9), it suffices to show that for every 1 #E q d 1, (5.10) q d 4 (#E) p 0 (#S j ) 1 1 p 0 q 3d d 1 p 0 Since p 0 and #S j q d 1, the inequality in (5.10) follows by a direct calculation. Thus the inequality in (5.5) holds. In order to prove the inequality in (5.6), just note from Corollary 15 that since q d 1 #E q d+1, we have 1 (#E) q d 1, {(x,y,z,s) E 4 :x+z=y+s} and then follow the same argument as in the proof of the inequality (5.5). The inequality in (5.7) follows from the inequalities in (5.5) and (5.6) because q 3d d 1 p 0 q 3d d 3 p 0, for p 0. Thus the proof of Lemma 16 is complete. From (3.3), recall that we have (5.11) Êdσ L (F d q,dm) q 1 E L (S j,dσ) for all characteristic functions E(x) on S j. Therefore Theorem 4 can be obtained by interpolating (5.11) and the inequalities in Lemma 16. References [1] A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra: cohomology and estimates, (1989), 130, [] J. Bourgain, N. Katz, and T. Tao, A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14, (004), [3] W. Duke and H. Iwaniec, A relation between cubic exponential and Kloosterman sums, Contemp. Math. 143, (1993), [4] J. Denef, and F. Loeser, Weights of exponential sums, intersection cohomology, and Newton polyhedra, Invent. Math. 106, (1991),

18 18 ALEX IOSEVICH AND DOOWON KOH [5] C.Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 14, (1970), [6] B. J. Green, Restriction and Kakeya phonomena, Lecture notes (003). [7] H. Iwaniec, and E. Kowalski, Analytc Number Theory, Colloquium Publications, 53, (004). [8] A. Iosevich and D. Koh, Erdös-Falconer distance problem, exponential sums, and Fourier analytic approach to incidence theorems in vector spaces over finite fields,(006), (preprint). [9] A. Iosevich and M. Rudnev, Erdos/Falconer distance problem in vector spaces over finite fields, TAMS (to appear), (007). [10] N. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Ann. Math. Studies, 116, Princeton (1988). [11] M. Lacey and W. McClain, On an argument of Shkredov in the finite field setting, (006), On-line journal of analytic combinatorics ( [1] R. Lidl and H. Niederreiter, Finite fields, Cambridge Univ. Press (1997). [13] G. Mockenhaupt, and T. Tao,Restriction and Kakeya phenomena for finite fields, Duke Math.J. 11, (004), [14] H, Salié, U ber die Kloostermanschen summen S(u,v;q), Math.Z.34, (193), [15] E.M.Stein, Harmonic Analysis, Princeton University Press, (1993). [16] R.S.Strichartz, Restriction of Fourier Transform to quadratic surfaces and decay of solutions of wave equations, Duke.Math.J., 44. (1977),no.3, [17] T.Tao, Recent progress on the restriction conjecture, (003), preprint. [18] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34, (1948), [19] A.Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50, (1974), Mathematics Department, 0 Mathematical Sciences Bldg, University of Missouri, Columbia, MO 6511 USA address: iosevich@math.missouri.edu Mathematics Department, 0 Mathematical Sciences Bldg, University of Missouri, Columbia, MO 6511 USA address: koh@math.missouri.edu