Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case

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1 ev. Mat. beroamericana 5 (009), no. 3, 7 68 Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case Shuanglin Shao Abstract For cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and bilinear adjoint restriction estimates. As corollaries, we first extend the ranges of exponents for the classical linear or bilinear adjoint restriction conjectures for such functions and verify the linear adjoint restriction conjecture for the paraboloid. We also interpret the restriction estimates in terms of solutions to the Schrödinger euation and establish the analogous results when the paraboloid is replaced by the lower third of the sphere.. ntroduction Let n 3 be a fixed integer and S be a smooth compact non-empty subset of the paraboloid { (τ,ξ) n : τ = ξ }.f0<p,, the classical linear adjoint restriction estimate for the paraboloid is the aprioriestimate (.) (gdσ) L ( n ) C p,,n,s g L p (S,dσ) for all Schwartz functions g on S, where (gdσ) (t, x) = g(τ,ξ)e i(x ξ+tτ) dσ(ξ) = g( ξ,ξ)e i(x ξ+t ξ ) dξ S n denotes the inverse space-time Fourier transform of the measure gdσ, and dσ is the canonical measure of the paraboloid defined in Section. 000 Mathematics Subject Classification: Primary 4B0, 4B5; Secondary 35Q55. Keywords: estriction conjecture, Schrödinger euation, cylindrical symmetry. n the notation of [7], the estimate (.) is denoted by S (p ) andtheestimate (.) is denoted by S,S (p p ).

2 8 S. Shao By duality, the estimate (.) is euivalent to the following estimate ˆf L p (S,dσ) C p,,n,s f L ( n ) for all Schwartz functions f, which roughly says that the Fourier transform of an L ( n ) function can be meaningfully restricted to the paraboloid S. This leads to the restriction problem, one of the central problems in harmonic analysis, which concerns the optimal range of exponents p and for which the estimate (.) should hold. t was originally proposed by Stein for the sphere [3] and then extended to smooth sub-manifolds of n with appropriate curvature [4] such as the paraboloid and the cone. The restriction problem is intricately related to other outstanding problems in analysis such as the Bochner-iesz conjecture, the local smoothing conjecture, the Kakeya set conjecture and the Kakeya maximal function conjecture, see e.g., [7], [8]. n this paper, we will mainly focus on the restriction estimates for the paraboloid. The corresponding linear adjoint restriction conjecture for the paraboloid asserts that Conjecture.. The ineuality (.) holds with constants depending on S, n and p, if and only if > n n+ and n. n p The conditions on p and are known to be best possible by the decay estimates of (dσ) and the standard Knapp example, see e.g., [4], [7]. When n =, the non-endpoint case was first proven to be true by Fefferman and Stein [7] (and generalized to other oscillatory integrals by Carleson and Sjölin [5]), and the endpoint case was proven to be true by Zygmund [9]. When n>, it was proven with the additional condition > (n+) by n Tomas [5] using real interpolation, and = (n+) by Stein [4] using complex interpolation. n 977, Córdoba [6] gave an alternate proof for n = n by largely relying on the successful resolution of the Kakeya conjecture in two dimensions. n 99, Bourgain [] generalized Córdoba s arguments to higher dimensions, so that nontrivial progress on the Kakeya problem might imply some nontrivial progress on the restriction result; using this techniue, he proved estimates for some < (n+) ; in particular, >4 when n =3. n 5 Further improvements along this line were made by Moyua, Vargas, Vega and Wolff, see e.g., [], [7]. The current best result > (n+) in higher n dimensions n 3 is due to Tao [0], based on the techniues in Wolff s breakthrough paper on the cone restriction estimates [8]. Among various techniues developed to attack this problem, the bilinear method proves to be very powerful. Variants of this idea also have applications to the nonlinear dispersive euations, see e.g., [], [3], [9], etc. More

3 Sharp linear and bilinear restriction estimates 9 precisely, we assume S and S to be two smooth compact non-empty subsets of the paraboloid in n, which are transverse in the sense that the unit normals of S and of S are separated by at least some fixed angle c>0. Then the classical bilinear adjoint restriction conjecture concerns the optimal range of exponents p and for which the bilinear operator, (f,g) (fdσ ) (gdσ ), should bound from L p L p to L,wheredσ,dσ are the canonical Lebesgue measures of S, S, respectively. The following formulation of this conjecture is taken from [3]. Conjecture.. Let S, S be defined as above and n, n+ + n n n p and n+ + n n. Then there exists a constant 0 <C< depending p on S,S,n and p, such that (.) (fdσ ) (gdσ ) L ( n ) C f L p (S ) g L p (S ) for all f L p (S ) and g L p (S ). t is known that the bilinear restriction conjecture. is stronger than the linear restriction conjecture., see [3]. For a discussion of recent progress made on this problem, see [7]. We remark that the conditions on p and in this conjecture are also known to be best possible by the decay estimates of (dσ) and the variants of the standard Knapp examples such as the suashed caps and the stretched caps, see e.g., [3], [7]. However, none of these Knapp-type examples are cylindrically symmetric functions, i.e., functions on n invariant under spatial rotations. Hence we expect that further estimates are available if we assume that functions are cylindrically symmetric and supported on a dyadic subset of the paraboloid in the form of {(τ,ξ) :M ξ M,τ = ξ } with M Z. We denote by L M this class of functions. ndeed, it is the case: when n = 3, the Tomas-Stein restriction estimate L L 4 is known to be best possible; but for functions in L M, the estimate L L is true for any > 0/3 by Corollary.3 in Section 3. Our main theorems, Theorem. and.5, of this paper are to present a family of sharp linear adjoint restriction estimates for f L, and bilinear ones for f L and g L M with 0 <M /4 on the dyadic space-time slab {/ x } with Z. The proofs essentially combine the two classical and elementary methods, the Carleson-Sjölin argument [4] and the bilinear method via the Whitney decomposition, which effectively solved the two dimensional restriction conjecture. n the arguments, we heavily exploit the rotational symmetry via the Fourier-Bessel formula, Lemma 3., for cylindrically symmetric functions to reduce matters to main term estimates by encoding the error term into certain integrals. A lot of effort is devoted to inventing counterexamples to show that the restriction

4 30 S. Shao estimates are best possible by relying on the idea coming from the standard Knapp examples, the principles of both stationary phase and non-stationary phase [4] and the Khintchine ineuality [7]. We remark that some of of them are uite challenging, see e.g., Example 4.5. As corollaries of the main theorems, we can verify the ineuality (.) for L M when the exponents p and are in a larger region (see Figure ) and show that it is nearly sharp except for certain endpoints. Furthermore, we show that the linear adjoint restriction conjecture. holds for all cylindrically symmetric functions when p and are restricted to the classical region. By similar arguments, one can also establish the analogous sharp restriction estimates when the paraboloid is replaced by the lower third of the sphere S n or more general cylindrically symmetric and compact hypersurfaces of elliptic type as defined in [], [3]. As applications of the restriction estimates, we will interpret them in terms of the solutions to the Schödinger euations and present another proof of the weighted Strichartz estimates in [6] for Schrödinger euations. Acknowledgements. The author is very grateful to his advisor Terence Tao for introducing this fascinating subject, and is indebted to him for many helpful conversations and encouragement during the preparation of this paper. The author thanks Monica Visan for helpful discussions. The author would like to thank the referee for his valuable comments and suggestions.. Notations and Main Theorems Let n 3 be the fixed space-time dimension. n this paper, we interpret n as the space-time freuency space. We will use the notations X Y, Y X, orx = O(Y )todenotethe estimate X CY for some constant 0 <C<, which may depend on p,, n and S or S, but not on the functions. f X Y and Y X we will write X Y. f the constant C depends on a special parameter other than the above, we shall denote it explicitly by subscripts. For example, C ε should be understood as a positive constant not only depending on p,, n and S or S, but also on ε. We denote by dσ the canonical Lebesgue measure of the standard paraboloid S = {(τ,ξ) :τ = ξ } in n, which is the pullback of n - dimensional Lebesgue measure dξ under the projection map (τ,ξ) ξ; thus, f(τ,ξ)dσ = f( ξ,ξ)dξ. S n By S n we denote the n dimensional unit sphere canonically embedded in n,andbydµ its surface measure.

5 Sharp linear and bilinear restriction estimates 3 We define a dyadic number to be any number Z of the form = j where j is an integer. For each dyadic number >0, we define the dyadic annulus in n, A := { x n : / x }. We define the space-time norm L t L r x of f on n by ( ( ) ) /r / f L t Lr x ( n ) = f(t, x) r dx dt, n with the usual modifications when or r are eual to infinity, or when the domain n is replaced by a small region of space-time such as A. When = r, we abbreviate it by L. Unless specified in this paper, all the space-time integrals are taken over A with dyadic >0, which will be clear from the context. We define the spatial Fourier transform of f on n by ˆf(ξ) = f(x)e ix ξ dx. n We use U to denote the indicator function of the set U, i.e., {, for x U, U (x) = 0, for x/ U. For p, we denote the conjugate exponent of p by p, i.e., /p + /p =. We start with stating the main theorem concerning the linear restriction estimates, which is proven in Section 3. Theorem.. Suppose f L and >0 is a dyadic number. Then the following sharp restriction estimates hold: for =and p, (fdσ) L min { n, for =4and 4 p, ε>0, (fdσ) L 4 ε min { n 4 +ε, n 4 } f L p (S). } f L p (S). for =3p and p<4, (fdσ) L min { } (n )( ), n f L p (S).

6 3 S. Shao for = and p, (fdσ) L min { n, } f L p (S). where A in {A, B} is given by the case where and B by. Furthermore, by interpolation we obtain the sharp restriction estimates L p L when p, are in the region determined by these lines (Figure ). L L 4 L 4 L 4 L p L with =3p and p<4 0 4 L L Figure : Linear restriction estimates on A. p emark.. We observe that the estimates above in each case are continuous in the sense that they match when. One can easily obtain the following corollary regarding the linear adjoint restriction conjecture. Corollary.3. Suppose f are cylindrically functions supported on the paraboloid. f f L M, the linear adjoint restriction conjecture. holds with constants depending on p,, n and M whenever > n, + n p and + n <n (Figure ). p The linear adjoint restriction conjecture. is true for all f (Figure ).

7 Sharp linear and bilinear restriction estimates 33 n n L n n L n n n (n ) L (n ) n L (n ) n 0 n n n (n ) L L p Figure : Linear restriction estimates for L M (cf. the inside trapezoid, the classical range of p and for Conjecture.). Proof. By Theorem., the first assertion follows from scaling to f L and then summing all dyadic and using interpolation. To prove the second assertion, it is sufficient to obtain it under the boundary conditions > n n+ and = n since other estimates are easily n p obtained by a standard argument of using the Hölder ineuality. By interpolating between the L L estimate and the L p L estimates on the line segment =3p and p<4, we obtain that, for > n, n+ = n n p and f L, (fdσ) L α() f L p (S), where α is the step function α() = { n n n [ ], for, (n ), for 0 <. We remark that the constant above C α() does not depend on ε. By scaling, when f L M, under the same conditions on p and, (fdσ) L (M)α(M) f L p (S), where α is defined as above. Then for all cylindrically symmetric f supported on the paraboloid, we decompose it into a sum of dyadically supported functions, f = f M, M: dyadic f {(τ,ξ):τ= ξ,m ξ M} = M

8 34 S. Shao where f M := f {(τ,ξ):τ= ξ,m ξ M}. Hence (fdσ) L ( n ) = ( (fdσ) L ( A ) ) / ( = ) / (f M dσ) L ( A ) M ( ( ) ) / (f M dσ) L ( A ) M ( ( ) ) / (M) α(m) f M L p (S) M ( /p f M p L (S)) = f p L p (S), M where >0 and M>0 are both dyadic numbers; in the last ineuality, since >0, M M >0, (M) α(m) <, (M) α(m) <, we have used the Schur s test for exponents p and satisfying the condition >p. > n n emark.4. n the cylindrically symmetric case, we remark that > n n is still sharp since it is given by the decay estimate (dσ) (t, ξ) C n ( + t + ξ ) ( n)/, see e.g., [4, Chapter 8, Theorem 3.]. Next we state the theorem regarding the bilinear restriction estimates in the cylindrically symmetric case, which is proven in Section 4. Theorem.5. Suppose f L and g L M with 0 <M /4. >0 is a dyadic number. Then the following sharp bilinear restriction estimates hold: for =and p, (fdσ ) (gdσ ) L min { M n n p, n + M n p, n n + } M p f L p (S ) g L p (S ).

9 Sharp linear and bilinear restriction estimates 35 for =and p, (fdσ ) (gdσ ) L min { n M n n p, for =4and 4 p, ε>0, n M p, n M n p } f L p (S ) g L p (S ). (fdσ ) (gdσ ) L 4 ε min { 3(n ) 4 +ε M n n p for =3p and p<4,, n 4 +ε M n p, n 4 M n p } f L p (S ) g L p (S ). (fdσ ) (gdσ ) L min { n M n n p for = and p,, (n )( ) M n p }, n M n p f L p (S ) g L p (S ). (fdσ ) (gdσ ) L min { (n ) M n n p, n n M p,m n p } f L p (S ) g L p (S ). where A in {A, B, C} is given by the case where /M,Bby /M and C by. Furthermore, by interpolation we obtain the sharp restriction estimates L p L p L when p and are in the region determined by these lines (Figure 3). L L L L L L V 4 L 4 L 4 L 4 V L p L p L with =3p and p<4 L L L 0 4 Figure 3: Bilinear restriction estimates on A. p

10 36 S. Shao emark.6. We observe that if and only if A B and if M and only if B C. n other words, the estimates are continuous in the sense that A B when /M and B C when. As a corollary of Theorem.5, we see that the bilinear adjoint restriction conjecture holds for exponents p and in a larger region. Corollary.7. Suppose f and g are defined as Theorem.5. Then the bilinear adjoint restriction conjecture. holds with constants depending on p,, n, S,S and M, whenever > n, + n <nand +. These n p p estimates are sharp except for certain endpoints (Figure 4). n n L n n n n L n L n L L L n n n n+ n n L L L n+ n L (n ) n L (n ) n L n n n n 0 n (n ) L L L p Figure 4: Bilinear restriction estimates for functions in L and L M with 0 <M /4 (cf. the inside pentagram, the classical range of p and for Conjecture.). emark.8. When S is replaced by the lower third of the sphere, the analogous results to Theorems. and.5 hold, which will be accomplished in Section 5. This is essentially due to the common geometric property of non-vanishing Gaussian curvature shared by the sphere and the paraboloid and the fact that the sphere locally resembles the paraboloid, which can be seen from the Taylor expansion ξ c( ξ ) ξ when ξ is small. emark.9. t is well known that the adjoint restriction estimates are closely related to the Strichartz estimates for the nonlinear dispersive euations such as the Schrödinger euation and the wave euation, see e.g., [6] and [4]. We will establish this connection in our case in Section 6.

11 Sharp linear and bilinear restriction estimates 37 This paper is organized as follows: Section 3 is devoted proving Theorem. and constructing counterexamples to show the linear estimates are sharp. Section 4 is devoted proving Theorem.5 and constructing counterexamples showing the bilinear estimates are sharp. n Section 5 we will establish analogous results for the cylindrically symmetric functions supported on the lower third of the sphere or the cylindrically symmetric and compact hypersurfaces of elliptic type. n Section 6 we will interpret the restriction estimates in terms of solutions to the Schrödinger euation to establish the Strichartz estimates. 3. Proof of Theorem.: linear estimates and examples For any cylindrically symmetric function f on the paraboloid, we set F ( ξ )= f( ξ,ξ). We observe that (fdσ) (t, x) is also a cylindrically symmetric function. To begin the proof of Theorem., we first investigate the behavior of (fdσ) on { x } via the following proposition. Proposition 3.. Suppose f L. Then for any p, max{,p } and, we have a sharp estimate (3.) (fdσ) L n f L p (S). Proof. f we change to polar coordinates, the left-hand side of (3.) is ( ) (fdσ) L = / f(ξ)e i(x ξ t ξ ) dξ dtdx / ξ ( / = F (s)s n e S its e irsω dµ(ω)ds dt r dr) n, n / where =[, ]. Then we change variables back s a / to majorize it by ( / a (n 3)/ F (a / )(dµ) (ra / e )e ita da dt r dr) n, / where =[, ] and e =(, 0,...,0) n. Then by the Hausdorff-Young ineuality when >orthe Plancherel theorem when =, changing s a = s and the fact that (dµ) L ω, the left-hand side of (3.) is further bounded by ( ) / n F dr n L / F () L (). Then by the Hölder ineuality and the fact F L p () f L p (S), (3.) follows.

12 38 S. Shao Now we will construct a counterexample to show the estimate (3.) is sharp when p and max{,p }.Wetake f( ξ,ξ)=f ( ξ ) = ξ (n ) { ξ } e it 0 ξ, where t 0. Then the left-hand side of (3.) reduces to ( / e i(t t 0)s S e irsω dµ(ω)ds dt r dr) n. n / We choose r and t satisfying that r [/00,/50], t t 0 /00. Then the left-hand side of (3.) n Hence (3.) is easily seen to be sharp. and its right-hand side n. Before investigating the behavior of (fdσ) on x, we shall exploit the spatial rotation-invariant symmetry of f in the following proposition. Lemma 3. (Fourier-Bessel formula). Suppose f is a cylindrically symmetric function supported on the paraboloid. Then (fdσ) (t, x) = c n r n + c n F (s)s n e its irs (3.) c n F (s)s n e i(rs ts ) ds + c n r n F (s)s n e its +irs 0 0 F (s)s n e i(rs+ts ) ds e rsy y n 4 [(y +i) n 4 (i) n 4 ]dyds e rsy y n 4 [(y i) n 4 ( i) n 4 ]dyds, where r = x and is the projection interval in the radial direction. Proof. We first expand (fdσ) in the polar coordinates, (fdσ) (t, x) = f( ξ,ξ)e i(xξ t ξ ) dξ = F (s)e its s n (dµ) (rse )ds, { ξ } where dµ is the surface measure of the sphere S n. On the one hand, the inverse Fourier transform of dµ is given by (dµ) (ξ) =c n ξ 3 n J n 3 ( ξ ), where J n 3 is the Bessel function of order n 3, see e.g. [4] or [5].

13 Sharp linear and bilinear restriction estimates 39 On the other hand, by using the same argument as proving [5, Lemma 3.], we obtain that, for fixed m 0, J m (r) =c m e ir e ir r / c m r m e ir 0 + c m r m e ir e ry y m 0 e ry y m [ ] (y +i) m (i) m dy [ (y i) m ( i) m ] dy. Hence (3.) holds after we combine these two estimates and set m= n 3. Therefore we define the error term of (fdσ) by Ef(t, x) :=± c n F (s)s n e its irs 0 e rsy y n 4 [(y ± i) n 4 (±i) n 4 ]dyds, where by ± we denote a sum of two terms where + and appear alternatively. Heuristically, one should think of Ef as a term comparable to r n/ n 4 F (s)s e its ds, which comes from estimating the error term of Bessel function J m (r) byr 3/. At the first approximation, this simplification is easy to deal with and intuitively provides what the bound involving the error term should be. However in this paper we will establish it rigorously in the following proposition, which shows that the information about its contribution to the linear estimates when x. t is acceptable compared to the main term estimates established in the next propositions. Proposition 3.3. Suppose f L. Then for any max{,p } and any dyadic number and f L p (S), (3.3) Ef L Proof. We set E(r) := 0 n + n f L p (S). e ry y n 4 [(y ± i) n 4 (±i) n 4 ]dy. For r, we first estimate E(r) by repeating the proof of [5, Lemma 3.] for readers convenience. E(r) e ry y n 4 (y ± i) n 4 (±i) n 4 dy+ 0 + =: + e ry y n 4 (y +i) n 4 (i) n 4 dy

14 40 S. Shao For, where0 y, by the mean value theorem we have (y ± i) n 4 (±i) n 4 y. For,wherey, we take y out and then use the mean value theorem to obtain (y ± i) n 4 (±i) n 4 y n 4. Then combining these estimates above, E(r) 0 e ry y n dy + e ry y n 4 dy r r n/ e y y n dy + r (n 3) e y y n 4 dy. By the definition of the Gamma function Γ, 0 r r n/ e y y n dy Γ(n/)r n/. 0 Then using integration by parts n 4 times when n 4andy r when n =3,weobtainr (n 3) e y y n 4 dy e r r. r Since e r r + n is continuous on [, ) and decays to 0 as r, e r r r n holds for r. Therefore (3.4) E(r) r n/. Next let us turn to the estimate (3.3). By changing to polar coordinates, the left-hand side of (3.3) is comparable to (3.5) ( / / F (s)s n e its irs E(rs) ds dt r dr) n, where =[, ]. After changing variables s = a /, (3.5) is comparable to ( / F (a / )a n 3 E(ra / )e ira/ e ita da dt r dr) n, / where =[, ]. Then by the Hausdorff-Young ineuality when > or the Plancherel theorem when =, changing variables back a = s and s, the left-hand side of (3.5) is further majorized by ( / / F (s)e(rs) ds dt r dr) n. / Since rs for any r [/,]ands, (3.4) and the Hölder ineuality give n + n F L p (). Since F L p () f L p (S), (3.3) follows. r

15 Sharp linear and bilinear restriction estimates 4 When x, we are left with estimating the main term of (fdσ), (3.6) Mf(t, x) :=c n r n F (s)s n e i(±rs ts ) ds, where by ± we denote the summation of two terms. We call it the heuristic approximation of (fdσ). We are going to prove the positive part estimates of Theorem. in the following four propositions. n the remainder of this section, we will prove its negative part sharpness by certain counterexamples. Proposition 3.4 ( = line). Suppose f L. Then for =, p and, we have a sharp estimate (3.7) (fdσ) L / f L p (S). Proof. We observe that it is sufficient to estimate the term of Mf with + sign. n the propositions followed, we will make the same reduction unless specified. Hence by the heuristic approximation (3.6) of (fdσ) with + sign, changing variables and then the Plancherel theorem in t followed by the Hölder ineuality, ( (fdσ) L = = ( / x / ( / / n r ) / (fdσ) (t, x) dtdx F (s)s n e i[rs ts ] ds F (s)s n e irs e ts ds dt dr dt r n dr ) / ) / ( ) / = F L ()dr / f s L (S) / f L p (S), where =[, ], p. Hence (3.7) follows. Now let us deal with the estimates on the line = 4. The estimate L 4 L 4 is the endpoint of two dimensional (n = ) linear adjoint restriction conjecture and hence the classical TT approach, namely the Carleson-Sjölin argument used in Proposition 3.6, unfortunately fails because we can not apply the Hardy-Littlewood-Sobolev ineuality at one step. nstead, we can perform a Whitney-type decomposition to to create some freuency separation. Similar arguments can be found in [0], [], [3].

16 4 S. Shao Proposition 3.5 ( = 4 line). Suppose f L. Then for =4, 4 p and ε>0 and, we have a sharp estimate up to ε, (3.8) (fdσ) L 4 ε (n )/4+ε f L p (S). Proof. By the same reasoning as that used in the proof of Proposition 3.4, we will only prove the estimate when p = 4. By the heuristic approximation (3.6) of (fdσ) with + sign, (fdσ) L 4 (n )/ ( / F (s)s n e i(rs ts ) ds 4 dt dr) /, where =[, ]. We set (Fdσ) (t, r) := n F (s)s e i[rs ts] ds, which can be regarded as the inverse space-time Fourier transform of f(τ,ξ) ξ n restricted to the parabola. To prove (3.8), it suffices to prove that, for any ε>0, (3.9) ( / ) / (Fdσ) (Fdσ) dtdr ε ε F L 4. Next we will perform a Whitney-type decomposition to =[, ]. For each j 0webreakup into O( j )dyadicintervalsτ j k of length j,andwrite τ j k τ j k if τ j k and τ j k are not adjacent but have adjacent parents. For each j 0, let F = F j k where F j k = F τ j k.then (Fdσ) (Fdσ) = j k,k :τ j k τ j k (F j k dσ) (F j k dσ). From the triangle ineuality, the left-hand side of (3.9) is bounded by (F j k dσ) (F j k dσ) j k,k :τ j k τ j L t,r ( )+ k + (F j k dσ) (F j k dσ) L =: A + B. j t,r ( A ) k,k :τ j k τ j k We will first estimate A. By the uasi-orthogonality property of functions among (F j k dσ) (F j k dσ) [3, Lemma 6.], ( ) /. A j k,k :τ j k τ j k (F j k dσ) (F j k dσ) L t,r ( )

17 Sharp linear and bilinear restriction estimates 43 By using the Plancherel theorem and the Cauchy-Schwarz ineuality and s i fori =,, (F j k dσ) (F j k dσ) L t,r ( ) F j k L ( j k ) F j k L ( j k ) dσj k dσj k L, where dσ j k and dσj k are two arc measures of the parabola {τ = ξ } in supported on τ j k and τ j k, respectively. On the one hand, from the geometric properties of the paraboloid, dσ j k dσj k L j. On the other hand, by the Hölder ineuality, F j k L ( j k ) j/4 F j k L 4 ( j k ). Thus after combining these estimates, ( ) /. A j k,k :τ j k τ j k F j k L 4 ( j k ) F j k L 4 ( j k ) We also observe that for each k, thereareo() k such that τ j k τ j k. Hence by the Cauchy-Schwarz ineuality, for any ε>0, A (log ) F L 4 ε ε F L 4. Next let us estimate B. On the one hand, by the Cauchy-Schwarz ineuality, (F j k dσ) L j/ F j k L (τ j k ). On the other hand, by the Plancherel theorem in t, ( ) / (F j k dσ) L t,r ( A ) = (F j k dσ) dt dr / F j k L (τ j ). k / Since there are O() k such that τ j k τ j k for each k, byusingthecauchy- Schwarz ineuality, B / j/ F j k L j (τ ) F j j k,k :τ j k τ j k k L (τ j k ) / j/ F L. k j Thus summing in j and using the Hölder ineuality, (3.9) follows. n contrast to the proof of the estimate L 4 L 4 in Proposition 3.5, the estimates L p L when =3p and p<4canbeprovenbythe Carleson-Sjölin argument or euivalently the TT method. Such arguments can also be used to prove the non-endpoint Strichartz estimates as in [8].

18 44 S. Shao Proposition 3.6 ( =3p line). Suppose f L. Then for p<4, =3p and, we have a sharp estimate (3.0) (fdσ) L (n )(/ /) f L p (S). Proof. By the heuristic approximation (3.6) with + sign, (fdσ) L (n )( ) ( where =[, ]. Setting (Fdσ) (t, r) := prove (3.) ( / / F (s)s n e i(rs ts ) ds dt dr) /, F (s)s n e i(rs ts) ds, we see that it suffices to F (s)s n e i(rs ts ) / ds dt dr) F L p (S). Suaring (Fdσ),weobtain {(Fdσ) (t, r)} = F (s )F (s )(s s ) n e i(r (s +s ) t(s +s )) ds ds, which is an oscillatory integral with a phase function r(s +s ) t(s +s ). ts Hessian is (s s ) which vanishes when s = s. But we can make a change of variables (s,s ) (a, b) witha = s + s,b= s + s. t is easy to see that the Jacobian is (s s ). Let Ω be the image in of under such change of variables. Then {(Fdσ) (t, r)} = F (a, b)e i(ra tb) da db, Ω where F (a, b) =F (s )F (s )(s s ) n / s s is a function of s and s. Setting =r. Since r >, by the Hausdorff-Young ineuality and s i fori =,, (Fdσ) (t, r) L ( A ) {(Fdσ) (t, r)} L r ( ) ( ) /r F (s,s ) r ds ds Ω ( ) /r F (s ) r F (s ) r s s ds ds r ( = F (s ) r F (s ) r s s ds ds ( r) ( ) /r F (s ) r r ( F r )(s )ds, ) /r

19 Sharp linear and bilinear restriction estimates 45 where r is the iesz potential of order r defined via the spatial Fourier transform by Îsf(ξ) = ξ s ˆf(ξ). Since f L p F L p (), it then suffices to prove that ( ) /r F r r ( F r )ds F L (). p By the Hölder ineuality, we obtain F r r ( F r )ds F r r L ( F r ). p/r () L /( r/p) () Since F r /r = F L p/r () L p (), it suffices to show r ( F r ) L /( r/p) F r L, which will follow from the Hardy-Littlewood-Sobolev ineuality. Hence the ineuality (3.) follows. p/r () Proposition 3.7 ( = line). Suppose f L. Then for =, p and, we have a sharp estimate (3.) (fdσ) L (n )/ f L p (S). Proof. By the heuristic approximation (3.6) of (fdσ) with + sign and the Hölder ineuality, for any p, (fdσ) L (n )/ F (s)s n e i(rs ts ) ds (n )/ f L L p (S). t,r Now we see that the restriction estimates in Theorem. follow from Propositions 3.3, 3.4, 3.5, 3.6 and 3.7. The remainder of this section is devoted constructing counterexamples. n view of the propositions above, we will use the main term of (fdσ), Mf(t, x) =c n r n F (s)s n e i(±rs ts ) ds, since the bound B given by the error terms are much smaller than that by the main terms when is sufficiently large. Our first counterexample is of Knapp-type, which is designed to show the estimates in the region in Figure determined by the estimates L L, L 4 L 4 and L L are sharp. The strength of the standard Knapp example or its variants lie in the idea of using both spatial localization and freuency localization. n this example, we will only show that the estimate L L is sharp since the computations for others are similar.

20 46 S. Shao Example 3.8 (). f, the L L estimate goes back to (3.7) in Proposition 3.4. We take f( ξ,ξ)=f ( ξ ) = ξ (n )/ { ξ + / }e ir 0 ξ +it 0 ξ, where r 0 [/,]andt 0. Thus the left-hand side of (3.7) is comparable to + / / e [ i(t t 0)s +i(±r r 0 )s] ds dt dr, / where by ± it denotes a summation of two terms. We observe that + / e [ i(t t 0)s +i(±r r 0 )s] ds + / = e i{(t t 0)(s ) [(±r r 0 ) (t t 0 )](s )} ds, and hence choose r [/,]andt such that /00 t t 0 /50, (r r 0 ) (t t 0 )](s ) / /00. Thus r and t are in the intersection region of two tubes whose size is of /. With this choice of r and t, (t t 0 )(s ) [(r r 0 ) (t t 0 )](s ) is less than a small number, say π/6. Then by direct computations, the term with + sign will be bounded below by /4. However for the term with sign, given this choice of r and t, we see that the roots of the uadratic polynomial, (t t 0 )(s ) [(r + r 0 )+(t t 0 )](s ), will be strictly less than, which conseuently are not located in the interval [0, / ]. Thus by the principle of non-stationary phase, we see that the term with sign will be bounded above by O N ( N ) for any N>0. Then by choosing N sufficiently large, from the triangle ineuality the left-hand side of (3.7) /4. Also it is easy to see that its right-hand side /4.Thuswesee that the estimate L L when issharp. Our second counterexample is to show that the estimates in the region in Figure determined by the lines =and = 4 are sharp. n this example we will only show the estimates on the line = in Proposition 3.4 are sharp by using the principle of stationary phase.

21 Sharp linear and bilinear restriction estimates 47 Example 3.9 (). f, the estimate L p L when p goes back to Proposition 3.4. We take the example f( ξ,ξ)=f ( ξ ) = ξ n { ξ } e ir 0 ξ +it 0 ξ, where r 0 [/,]andt 0. Then the left-hand side of (3.7) is comparable to ( / e i{(t t 0)(s ) [(±r r 0 ) (t t 0 )](s )} ds dt dr). / We choose r [/00,/50] and t such that (r r 0 )/(t t 0 ) [, ]. Then [(r r 0) (t t 0 )] (t t 0 ) [0, ], (r + r 0)+(t t 0 ) (t t 0 ) <. Then from the principles of stationary phase and non-stationary phase, e i{(t t 0)(s ) [(r r 0 ) (t t 0 )](s )} ds /, e i{(t t 0)(s ) +[(r+r 0 )+(t t 0 )](s )} ds N N, for any N 0. Then if choosing N sufficiently large, the triangle ineuality gives ( / e i{(t t 0)(s ) [(±r r 0 ) (t t 0 )](s )} ds dt dr) /. / ts right-hand side / for p. Thus we see that the estimates on the line =when aresharp. The third counterexample shows that the estimates inside the region determined by lines =4, =3p and = in Figure are sharp. n this example, we will only carry out the computations for the estimates L p L on the line = in Proposition 3.7. Example 3.0 (). f, the estimate L p L when p goes back to Proposition 3.7. We take f( ξ,ξ)=f ( ξ ) = ξ n { ξ } e ir 0 ξ +it 0 ξ, where r 0 [/,]andt 0. They are chosen such that (fdσ) L can be realized at (t 0,x 0 )withr 0 = x 0.

22 48 S. Shao Hence the left-hand side of the ineuality (3.) is comparable to n e i(±r 0 r 0 )s ds. Since r 0 [/,], e ir 0s ds N holds for any N 0. Then the triangle ineuality yields n ei(±r 0 r 0 )s ds n. On the other hand, the right-hand side of (3.) (n )/ for p. Hence the estimates on the line = when aresharp. For lines =4and =3p, the estimates go back to (3.8) and (3.0). We choose r [/,]andt such that r r 0 4, t t 0 4. Then by the same reasoning as Example 3.8, these estimates are sharp. Thus the proof of Theorem. is complete. 4. Proof of Theorem.5: bilinear estimates and examples For f L and g L M with 0 <M /4, we set =[, ], M =[M,M] and F ( ξ ) =f( ξ,ξ),g( ξ ) =g( ξ,ξ). n the bilinear case, and /M will bring in two natural separation scales in the physical space. n light of the proof of Theorem., we will have the following permutations of the product (fdσ ) (gdσ ). when x = r /M, (fdσ ) (gdσ ) = MfMg + MfEg + MgEf + EfEg. when x = r /M, (fdσ ) (gdσ ) = Mf(gdσ ) + Ef(gdσ ). when x, (fdσ ) (gdσ ) remains unchanged. We are going to prove the estimates part of Theorem.5 via the following three propositions and its sharpness part by building counterexamples in three cases followed. Proposition 4.. Suppose f L and g L M with 0 <M /4, and is a dyadic number. Then we have sharp estimates for =and p, (4.) (fdσ ) (gdσ ) L n n + M p f L p (S ) g L p (S ). for max{,p }, (4.) (fdσ ) (gdσ ) L n M n p f L p (S ) g L p (S ).

23 Sharp linear and bilinear restriction estimates 49 Proof. f we change to the polar coordinates, the left-hand side of (4.) reduces to F (s )e its (dµ) (rs e )s n ds / (4.3) G(s )e its (dµ) (rs e )s n ds dtr n dr. M We use the Cauchy-Schwarz ineuality and the Plancherel theorem in t to bound (4.3) by (4.4) n M n F L ( )M / G L ( M ). Then by the Hölder ineuality, (4.4) is bounded by n n n M p f L p (S ) g L p (S ). Hence the ineuality (4.) follows. To prove (4.), by the Hölder ineuality, (fdσ ) (gdσ ) L (fdσ ) L (gdσ ) L. On the one hand, by Proposition 3., (fdσ ) n L f L p (S ). On the other hand, by the Hölder ineuality, Hence the ineuality (4.) follows. (gdσ ) L M n p g L p (S ). The following proposition concerns the case where x /M. Proposition 4.. Suppose f L and g L M with 0 <M /4, and /M.Then for =and p, (4.5) (fdσ ) (gdσ ) L n M + n p f L p (S ) g L p (S ). for max{,p }, (4.6) (fdσ ) (gdσ ) L n L p LM p f L p (S ) g L p (S ), where L p L denotes the operator norm of f (fdσ) from L p (S ) to L given by Theorem..

24 50 S. Shao Proof. To prove (4.5), it suffices to prove the following ineualities by Lemma 3., Mf(t, re ) (gdσ ) (t, re ) n dt r dr n + M n p f g, L p (S ) L p (S ) / and / Ef(t, re ) (gdσ ) (t, re ) dt r n dr n M + n p f L p (S ) g L p (S ). These two estimates above can be proven along similar lines as proving (4.). We choose to prove the second. The Hölder ineuality yields, (fdσ ) (gdσ ) L (fdσ ) L (gdσ ) L. Then by using the same reasoning as in the proof of (4.), the ineuality (4.6) follows. WenotethattheerrortermEf givesabetterdecayestimateas expected. Next let us concentrate on the case where x /M. As indicated at the beginning of this section, we will have to deal with estimates involving MfMg, MfEg, EfMg and EfEg. Proposition 4.3 (Bilinear main term estimates). Suppose f L g L M with 0 <M /4, and /M.Then for =and p, (4.7) (fdσ ) (gdσ ) L M n n p f L p (S ) g L p (S ). for =and p, (4.8) (fdσ ) (gdσ ) L n M n n p f L p (S ) g L p (S ). for max{4, 3p } and p, (4.9) (fdσ ) (gdσ ) n n L L p L M n p f L p (S ) g L p (S ). Proof. To prove (4.7), it suffices to prove the following ineualities and (4.0) MfMg L M n n p f L p (S ) g L p (S ). EfEg L M n 4 n p f L p (S ) g L p (S ). MfEg L M n 4 n p f L p (S ) g L p (S ). EfMg L M n n p f L p (S ) g L p (S ).

25 Sharp linear and bilinear restriction estimates 5 n what follows we will only prove (4.0) since other estimates involving error terms will follow similarly. n fact these ineualities give better decay estimates than those given by (4.0). By the heuristic approximation (3.6) with + sign, the left-hand side of (4.0) reduces to (4.) / F (s )s n e i(rs ts ) ds G(s )s n e i(rs ts ) dt dr. M After changing variables and using the Cauchy-Schwarz ineuality and the Plancherel theorem in t, we see that (4.) is bounded by M n 3 F L ( ) G L ( M ) M f L (S ) g L (S ). Then from the Hölder ineuality, the ineuality (4.0) follows. Similarly, to prove (4.8), it suffices to prove the following ineualities (4.) MfMg L n M n n p f L p (S ) g L p (S ). EfEg L n+ M n n p f L p (S ) g L p (S ). MfEg L n + n M n p f L p (S ) g L p (S ). EfMg L n + n M n p f L p (S ) g L p (S ). Since the estimates above involving error terms give better decay estimates than (4.), we will also only prove (4.). We rewrite its left-hand side as ( n F (s )G(s ) M / (s s ) n e i(r(s +s ) t(s +s ) ds ds dtdr ) /. Setting x := s + s and y := s + s, we observe that the Jacobian M providedm /4. From the Plancherel theorem both in t and r, the left-hand side of (4.) is further majorized by n M n F L ( ) G L ( M ) n f L (S ) g L (S ). By using the Hölder ineuality again, we see that (4.) follows. Finally we prove (4.9). n fact, it suffices to prove the following two ineualities (fdσ ) Mg L n n L p L M n p f L p (S ) g L p (S ). (fdσ ) Eg L L p L n n n M p f L p (S ) g L p (S ). The first follows from the Hölder ineuality and the linear estimate in Theorem., and the second follows along similar lines.

26 5 S. Shao Therefore the restriction estimates in Theorem.5 are obtained from Propositions 4., 4. and 4.3. n the remainder of this section we will construct counterexamples to show these estimates are sharp or nearly sharp up to ε. Since the error terms give much better decay estimates, we will use the heuristic approximations (3.6) of (fdσ ) and (gdσ ) when computing these examples. We will distinguish them into three cases as follows. Case : /M. We start with a common example to show the estimates in the region in Figure 3 determined by L L L, L L L and L L L are sharp by using the idea coming from standard Knapp example. n the following example, we will only do the computations when p =and =. Example 4.4 (). f /M, M ineuality is best possible in the following (4.3) (fdσ ) (gdσ ) L M f L (S ) g L (S ). We take f( ξ,ξ)=f( ξ ) = ξ n e i( r 0 ξ +t 0 ξ ) { ξ + M}, g( η,η)=g( η ) = η n e i( r 0 η +t 0 η ) {M η M+ }, where r 0 [/,]andt 0. By the heuristic approximation (3.6), the left-hand side of (4.3) is comparable to / + M M+ e i[(±r r 0)s (t t 0 )s ] ds e i[(±r r 0)s (t t 0 )s ] ds dt dr, which we understood is a summation of four terms. For the integral on [, + M], we will choose r and t such that /00 r r 0 /50 and M /00 t t 0 M /50. Then we have [(r r 0 ) (t t 0 )](s ) (t t 0 )(s ), (r + r ( 0)+(t t 0 ) = + r r ) 0 / [0, M]. (t t 0 ) (t t 0 ) Hence the integral on [, + M] with + sign is M while the one with sign N N for any N 0. Similarly for the integral on M

27 Sharp linear and bilinear restriction estimates 53 [M,M + ] with this choice of r and t. ThenifN is sufficiently large, the triangle ineuality gives + M e i[(±r r 0)s (t t 0 )s ] ds / M e i[(±r r 0)s (t t 0 )s ] ds dt dr. M Then by direct computations, the right-hand side of (4.3). Thus we see that the estimate L L L is sharp when /M. By modifying the above narrow Example 4.4, namely taking a linear combination to create a spreading-out example, we will show that the estimates in the region in Figure 3 determined by the lines =and = are sharp by using the Khintchine ineuality. A similar construction by Lee and Vargas can be found in [0] to show the sharp null form estimates for the wave euation. n the following example we will only do computations for the estimates on the line =. Example 4.5 (). f /M, M (n )/ (n )/p is best possible in the following ineuality (4.4) (fdσ ) (gdσ ) L M n n p f L p (S ) g L p (S ), where p. We define two index sets J := {j Z : j [M ]} and K := {k Z : k [M]}, where[x] denotes the biggest integer which is less than or eual to x. Foreachj J, k K we define f j ( ξ,ξ)=f j ( ξ ) = ξ n e i( r 0 ξ +t 0 ξ ) {+(j ) M ξ +j M}, g k ( η,η)=g k ( η ) = η n e i( r 0 η +t 0 η ) {M+(k ) η M+k }. Also we set f = j J ε jf j and g = k K ε kg k,where{ε j : j J} and { ε k : k K} are sets of i.i.d. (independent identically distributed) random variables taking ± with an eual probability /. Note that f j and g k are narrow, disjoint and in the form of Example 4.4; but f and g spread out and support on the whole set S and S. By the Khintchine ineuality, we estimate the left-hand side of (4.4) by ) (4.5) E ( (fdσ ) (gdσ ) L ( (f j dσ ) (g k dσ ) ) / L, j,k

28 54 S. Shao where E(X) denotes the expectation of the random variable X. Bythe heuristic approximation (3.6), the right-hand side of (4.5) is comparable to ( aj F j (s )e i[(±r r 0)s (t t 0 )s ] ds / j,k a j M (4.6) bk ) G k (s )e i[(±r r 0)s (t t / 0 )s ] ds dt dr, b k where a j =+j M and b k = M +k,andby± we denote a summation of four terms. We choose r and t such that /00 r r 0 /50 and M t t 0 M /50. By this choice of r and t and similar discussions as in Example 4.4, the triangle ineuality gives, aj F j (s )e i[(±r r 0)s (t t 0 )s ] ds a j M bk G k (s )e i[(±r r 0)s (t t 0 )s ] ds M. b k Then (4.6) is bounded below by M ( J K ) / M, i.e., ( J K ) /. Here J M denotes the cardinality of the index set J, similarly for K M. Hence we obtain that the left-hand side of (4.4) is. On the other hand, the right-hand side of (4.4) for p. Hence the estimates on the line =when /M are sharp. The following example shows that the estimates in the region in Figure 3 determined by the lines =and = 4 are sharp by the principle of stationary phase. We do computations when =. Example 4.6 (). f /M, n M n n p is best possible in the following ineuality. (4.7) (fdσ ) (gdσ ) L n n M n p f L p (S ) g L p (S ), where p. We take f( ξ,ξ)= ξ n e i( r 0 ξ +t 0 ξ ) { ξ }, g( η,η)= η n e i( r 0 η +t 0 η ) {M η M}, where r 0 [/,]andt 0. By the heuristic approximation (3.6), the left-hand side of (4.7) is comparable to ( / n e i[(±r r 0)s (t t 0 )s ] ds e i[(±r r 0)s (t t 0 )s ] ds dt dr). M /

29 Sharp linear and bilinear restriction estimates 55 We choose r [/,]andt such that M /00 r r 0 M /50 and (r r 0 )/(t t 0 ). Then from the principles of stationary phase and non-stationary phase, for any N 0, e i[(r r 0)s (t t 0 )s ] ds M /, e i[(r+r 0)s +(t t 0 )s ] ds N M N, e i[(r r 0)s (t t 0 )s ] ds M, s M e i[(r+r 0)s +(t t 0 )s ] ds N M N. s M Then from the triangle ineuality, the left-hand side of (4.7) (n ) M /. By direct computations, the right-hand side of (4.7) (n ) M /.Thus the estimates on the line =when /M are sharp. The next example will show that the estimates in the region V in Figure 3 determined by L L L, L 4 L 4 L 4 and L L L are sharp up to ε by using the idea of Knapp example. We will only do computations for the estimate when p = =. Example 4.7 (V). f /M, n is best possible in the following ineuality. (4.8) (fdσ ) (gdσ ) L n f L (S ) g L (S ). We take f( ξ,ξ)= ξ n e i( r 0 ξ +t 0 ξ ) { ξ +M }, / g( η,η)= η n e i( r 0 η +t 0 η ) {M η M}, where r 0 [/,]andt 0. By the heuristic approximation (3.6), the left-hand side of (4.8) is comparable to ( +M ) n e i[(±r r 0)s (t t 0 )s ] ds e i[(±r r 0)s (t t / 0 )s ] ds dt dr. / M We choose r [/,]andt such that (r r 0 ) (t t 0 ) M /, (r r 0 ) M(t t 0 ) M, M /00 t t 0 M /50,

30 56 S. Shao i.e., r and t are located in the intersection area of two tubes which has size M M /. Then by similar discussions as Example 4.4, the left-hand side of (4.8) (n )/ M 3/4 ; by direct computations the right-hand side of (4.8) (n )/ M 3/4. Hence we see that the estimate L L L when /M is sharp. The next example shows that the estimates in the region V in Figure 3 determined by the lines = 4, = and = 3p are sharp. n this example, we will do the computations for the estimates on the line =. Example 4.8 (V). f /M, (n ) M n n p following ineuality is best possible in the (4.9) (fdσ ) (gdσ ) L (n ) M n n p f L p (S ) g L p (S ), where p. We take f( ξ,ξ)= ξ n e i( r 0 ξ +t 0 ξ ) { ξ }, g( η,η)= η n e i( r 0 η +t 0 η ) {M η M}, where r 0 [/,]andt 0 satisfying the L norms (fdσ) L ( A ) and (gdσ) L ( A ) can be realized at (t 0,x 0 )with x 0 = r 0. By the heuristic approximation (3.6), the left-hand side of (4.9) is comparable to (n ) e i(±r 0 r 0 )s ds e i(±r 0 r 0 )s ds. M Then by the same reasoning as Example 3.0, the above (n ) M. On the other hand, the right-hand side of (4.9) (n ) M. Hence the estimates on the line = when /M are sharp. For lines =4, 4 p or =3p, p<4, the estimates go back to (4.9). We will choose r [/,]andt such that r r 0 4and t t 0 4. Then by similar reasoning, the estimates on these lines are sharp. Case : /M. n this subcase, we will construct counterexamples to show the restriction estimates in Theorem.5 are sharp when /M.AsintheCase, we will start with a narrow example which shows that estimates in the region in Figure 3 determined by L L L, L L L and L L L are sharp. n this example, we will do computations for the estimate L L L.

31 Sharp linear and bilinear restriction estimates 57 Example 4.9 (). f /M, n M n 3 is best possible in (4.0) (fdσ ) (gdσ ) L n M n 3 f L (S ) g L (S ). We take f( ξ,ξ)= ξ n e i( r 0 ξ +t 0 ξ ) { ξ +M }, g( η,η)= η (n ) e it 0 η M, where r 0 [/,]andt 0. By the heuristic approximation (3.6) only for (fdσ), we see that the left-hand side of (4.0) is comparable to +M n e i((±r r 0)s (t t 0 )s ) i(t t0)s ds (dµ) (rse )ds dt dr. / We choose r [/,]andt such that M e /00 r r 0 /50, M /00 t t 0 M /50. Then we have [(r r 0 ) (t t 0 )](s ) c with a small c>0and <. From the principle of non-stationary phase and the (r+r 0)+(t t 0 ) (t t 0 ) triangle ineuality, the left-hand side of (4.0) n M. On the other hand, the right-hand side of (4.0) n M. Thus the estimate L L L when /M is sharp. n the next example, we are going to show the estimates in the region in Figure 3 determined by the lines =and = are sharp. n this example, we will do computations for the estimates on the line =. Example 4.0 (). f /M, n n + M p following ineuality is best possible in the (4.) (fdσ ) (gdσ ) L n M + n p f L p (S ) g L p (S ), where p. We define an index set J := {j : j [M ]}. For each j J, weset f j ( ξ,ξ)=f j ( ξ ) = ξ n e i( r 0 ξ +t 0 ξ ) {+(j )M ξ +jm }. Then we define f = j ε j f j, g( η,η)= η (n ) e it 0 η M, where {ε j : j J} is a set of i.i.d. random variables taking ± withaneual probability /.

32 58 S. Shao By using the Khintchine ineuality, we obtain ( ) E (fdσ ) (gdσ ) L ( A ) ( (f j dσ ) (gdσ ) ) / L ( A ), j where E(X) denotes the expectation of the random variable X. By the heuristic approximation (3.6), the right-hand side of the above is comparable to ( cj n e i[(±r r 0)s (t t 0 )s ] ds c j M / j ) i(t t / 0)s e (dµ) (rs e )ds dt dr, M where c j =+jm.wechooser [/,]andt such that /00 r r 0 /50, M /00 t t 0 M /50. This gives [(r r 0 )(t t 0 )](s c j ) c and (r+r 0)+(t t 0 ) (t t 0 <. Then by ) the principle of non-stationary phase and the triangle ineuality, the above is bounded below by M J / M 3,where J M denotes the cardinality of the set J. Hence the left-hand side of (4.) n/ and the right-hand side of (4.) n/ for p. Thus the estimates on the line = when /M are sharp. n the following example, we will see the estimates in the region in Figure 3 determined by the lines =and = 4 are sharp in the case /M. We will do computations for the estimates on the line = below. Example 4. (). f /M, / M (n )/p is best possible in the following ineuality. (4.) (fdσ ) (gdσ ) L / M (n )/p f L p (S ) g L p (S ), where p. We take f( ξ,ξ)= ξ n e i( r 0 ξ +t 0 ξ ) { ξ }, g( η,η)= η (n ) e it 0 η M, where r 0 [/,]andt 0. Then by the heuristic approximation (3.6), the left-hand side of (4.) is comparable to ( e i[(±r r 0)s (t t 0 )s ] i(t t / 0)s ds e (dµ) (rs e )ds dt dr). M /

33 Sharp linear and bilinear restriction estimates 59 r r We choose r [/00,/50] and t such that 0 (t t 0 ). Then r and t are in the region of size. The principles of stationary phase and non-stationary phase again give, for any N 0, e i[(r r 0)s (t t 0 )s ] ds, e i[(r+r 0)s +(t t 0 )s ] ds N N. With this choice of r and t, wehave M e i(t t 0)s (dµ) (irs ω)ds M. Hence from the triangle ineuality, the left-hand side of (4.) / M. On the other hand, the right-hand side of (4.) / M. Thus we see that the estimates on the line =when /M are sharp. The next example will show that the estimates in the region V in Figure 3 determined by L L L, L 4 L 4 L 4 and L L L are sharp. We will do the computations for the estimate L L L. Example 4. (V). f /M, / M (n )/ is best possible in the following ineuality. (4.3) (fdσ ) (gdσ ) L M n f L (S ) g L (S ). We take f( ξ,ξ)= ξ n e i( r 0 ξ +t 0 ξ ) { ξ + }, / g( η,η)= η (n ) e it 0 η M, where r 0 [/,]andt 0. By the heuristic approximation for (fdσ), the left-hand side of (4.3) is comparable to / + / e i[(±r r 0)s (t t 0 )s ] ds M e We choose r [/,]andt such that i(t t0)s (dµ) (irs e )ds dt dr (r r 0 ) (t t 0 ) / /00, / /00 t t 0 / /50. Then r and t are located in the intersection area of two tubes which has size of /. Hence the left-hand side of (4.3) /4 M. On the other hand, its right-hand side /4 M. Thus we see that the estimate L L L when /M is sharp. The following example will show that the estimates in the region V in Figure 3 determined by the lines =4, = and =3p are sharp. /.

34 60 S. Shao Example 4.3 (V). f /M, (n )/ M (n )/p in the following ineuality is best possible (4.4) (fdσ ) (gdσ ) L n M n p f L p (S ) g L p (S ), where p. We take f( ξ,ξ)= ξ n e i( r 0 ξ +t 0 ξ ) { ξ }, g( η,η)= η (n ) e it 0 η M, where r 0 [/,]andt 0. They are chosen such that (fdσ ) L and (gdσ ) L can be realized at (t 0,x 0 )with x 0 = r 0. By the heuristic approximation (3.6), n e i(±r 0 r 0 )s ds (dµ) (rs e )ds. M Then from the triangle ineuality, the left-hand side of (4.4) (n )/ M. On the other hand, its right-hand side (n )/ M for p.thus the estimates on the line = when /M are sharp. When =4, 4 p,or =3p, p<4, the estimates go back to (4.6). We choose r [/,]andt such that r r 0 4and t t 0 4. Case 3:. n this subcase, we will construct counterexamples to show the estimates (4.) and (4.) are sharp. We will omit the computations for simplicity. The following example shows that the estimates in the region determined by L L L, L L L and L L L are sharp. Example 4.4 (). We take f( ξ,ξ)=f( ξ ) = ξ (n ) e it 0 ξ { ξ +M }, g( η,η)=g( η ) = η (n ) e it 0 η M, where t 0. The r and t are chosen such that r and t t 0. 50M 00M The next example shows that the estimates in the region determined by the lines =and =aresharp.

35 Sharp linear and bilinear restriction estimates 6 Example 4.5 (). We define an index set J := {j : j [M ]}. For each j J, weset Then we define f j ( ξ,ξ)=f j ( ξ ) = ξ (n ) e it 0 ξ {+(j )M ξ +jm }. f = j ε j f j, g( η,η)= η (n ) e it 0 η M, where {ε j : j J} is a set of i.i.d. random variables taking ± withan eual probability /, and the r and t are chosen such that / r and / t t 0. The third example shows that the estimate (4.) is sharp. Hence the estimates in the regions, V and V when aresharp. Example 4.6 (, V, V). We take f( ξ,ξ)=f( ξ ) = ξ (n ) e it 0 ξ, g( η,η)=g( η ) = η (n ) e it 0 η M, where t 0. The r and t will be are chosen such that t t 0. Thus the proof of Theorem.5 is complete. r and 5. Connection with the restriction estimates for the sphere or the hypersurface of elliptic type n this section we are concerned with whether the analogous results of Theorems. and.5 remain valid if S is replaced with the lower third of the sphere S n or a cylindrically symmetric and compact hypersurface of elliptic type. Let us first consider the case where the paraboloid is replaced by the sphere S n in n. Suppose f is a cylindrically symmetric function supported on a compact set of S n, S := {( ξ,ξ) n : M ξ M}, where0<m /6. Then (5.) (fdµ) (t, x) = e i(x ξ t ξ ) F ( ξ )dξ, M ξ M where dµ is the surface measure of the sphere and F ( ξ ) =f( ξ,ξ).