Fourier restriction to convex surfaces of revolution in R3

Publ. Mat. 50 (2006), 71–85 FOURIER RESTRICTION TO CONVEX SURFACES OF REVOLUTION IN R3 Faruk Abi-Khuzam and Bassam Shayya Abstract If Γ is a C3hypersurface in Rnand dσ is induced Lebesgue measure on Γ, then it is well known that a Tomas-Stein Fourier restriction estimate on Γ implies that Γ has a nowhere vanishing Gaussian curvature. In a recent paper, Carbery and Ziesler observed that if induced Lebesgue measure is replaced by affine surface area, then a Tomas-Stein restriction estimate on Γ implies that Γ satisfies the affine isoperimetric inequality. Since the only property needed for a hypersurface to satisfy the affine isoperimetric inequality is convexity, this raised the question of whether a TomasStein restriction estimate can be obtained for flat but convex hypersurfaces in Rnsuch as Γ(x) = (x, e−1/|x|m), m= 1,2, . . . . We prove that this is indeed the case in dimension n= 3. 1. Introduction Let Γ be a C3hypersurface in Rnand dσ a measure on Γ. A TomasStein Fourier restriction estimate for the pair (Γ, dσ) is an inequality of the form (1) kb fkL2(dσ).kfkL 2n+2 n+3 (Rn) for f∈C0(Rn). The existence of restriction estimates such as (1), as well as their connection with the geometry of Γ, or with the decay of the Fourier transform of dσ, has been a subject of great interest. See [9, pp. 368–373] for some important applications of these estimates. The choice of the measure dσ is not completely arbitrary. It usually reflects some aspect of the geometry of Γ. Two important choices of dσ are induced Lebesgue measure and affine surface area. In the former case, if Γ is assumed to have non-vanishing Gaussian curvature, (1) is a 2000 Mathematics Subject Classification. 42B10, 42B15. Key words. Fourier transform, restriction, affine surface area. 72 F. Abi-Khuzam, B. Shayya classical result of Tomas and Stein (see [10] and [9]). Conversely, if (1) holds with induced Lebesgue measure, then a result of Iosevich and Lu [3] (see also [2]), implies that Γ has non-vanishing Gaussian curvature. The proof of this converse uses, among other things, a Knapp-type scaling argument. To see how this argument goes, consider the special case where Γ is a surface of revolution given by Γ(x) = (x, φ(x)), where φ(x)=γ(|x|), and γ: [0, b)→Ris increasing and satisfies γ(0)=γ′(0)=0. For 0 < δ < b, let Sδ={(x, γ(|x|) : |x| ≤ δ}and let fδbe a smoothedout characteristic function of Sδ. It is then easy to see that kfδkL2(dσ). δ(n−1)/2, and that |[ fδdσ|&δn−1on a (C/δ)× ··· × (C/δ)×(C/γ(δ)) box in Rn(for a suitable constant C). Now if (1) holds then, by duality, the equivalent adjoint restriction estimate (2) kd f dσkL 2n+2 n−1(Rn).kfkL2(dσ) also holds. Applying (2) to fδwe obtain (3) δ2.γ(δ) and this implies that γ′′(0) 6= 0. In particular γcannot have vanishing Gaussian curvature at the origin. A more elaborate argument shows that the same conclusion holds in general. In the latter case, say when Γ(x) = (x, φ(x)), the affine surface area on Γ is given as the pushforward under Γ of the (n−1)-dimensional measure |Kφ(x)|1/(n+1) dx, where Kφ(x) = det(Hess φ(x)) is the affine curvature of Γ. To see what kind of geometry on Γ may be expected, take the case of a surface of revolution considered above. The radial assumption on φ, e.g. φ(x) = γ(|x|), simplifies matters and one computes that Kφ(x) = γ′′(|x|)γ′(|x|) |x|n−2 . If we then take dσ in the adjoint restriction estimate (2), which is equivalent to (1), to be affine surface area and use the function fδin it, we arrive [1] at the inequality Zδ 0γ′′(r)γ′(r) rn−2 1/(n+1) rn−2dr .δn−1γ(δ)(n−1)/(n+1) . But now this inequality does not imply non-vanishing curvature. Rather, it is satisfied by any convex γ, regardless of how flat it is at the origin, e.g. it is satisfied by γ(t) = e−1/tm,many positive integer. In fact, even if φis not radial, there is a similar scaling argument that can be applied, and it leads to the conclusion that φsatisfies the affine isoperimetric Restriction to Convex Surfaces 73 inequality of affine differential geometry, which is certainly true whenever φis convex. For more details we refer the reader to [1, pp. 409–410], [5, Chapter 5], and [6]. An earlier result of Sj¨olin [8] had already established that, if the dimension n= 2, and φis convex, then the restriction inequality holds true for affine surface area. The strength of this result, along with the above considerations, suggested that, perhaps, the geometric condition of convexity of φcould imply a restriction result for affine surface area in higher dimensions. But if only convexity is to be used, functions such as φ(x) = e−1/|x|mhave to be admitted. In attempting to prove this result, i.e. to show that convexity implies restriction, Carbery and Ziesler [1] considered the implications of a decay assumption on the Fourier transform of dσ. Kenig, Ponce and Vega [4] proved that if the decay assumption (4) ZB(0,b) e−2πiξ·Γ(x)|Kφ(x)|1 2+iα dx.(1 + |α|)N |ξn| was true for all real αand some integer N, then (2) holds1. When testing (4) on φ(x) = e−1/|x|m, Carbery and Ziesler [1] found that it did not hold true in dimension n= 3. This, of course, did not mean that there was no restriction result for φ(x) = e−1/|x|m. More recently, the same restriction question was addressed in [7]. A consequence of the results there implies that if φ(·) = γ(| · |), where γis convex, γ(0) = γ′(0) = 0, γ(3)(t) non-negative, and if sup 0<t<b tγ′′(t) γ′(t)≤C < ∞, then the restriction estimate (1) holds for affine surface area in dimension n= 3. Testing this last condition on γ(t) = e−1/tm, where 0< t < bm,bm=m/(3m+ 3)), one finds that sup 0<t<bm tγ′′(t) γ′(t)= sup 0<t<bmm tm−m−1=∞. Once again, the function e−1/tmwas precluded from the result. It turns out that, at least for surfaces of revolution Γ(x) = (x, φ(x)), φ(x) = γ(|x|), a Tomas-Stein restriction estimate for affine surface area 1This connection between decay and restriction is valid in dimensions n= 2,3. In dimensions n≥4, one has to modify things slightly by inserting a smooth cut-off function into both (2) and (4), see [1] for further details. 74 F. Abi-Khuzam, B. Shayya does hold in the presence of convexity, if we add the condition that (5) sup 0<t<b γ(t)γ′′(t) γ′(t)2≤C < ∞. Now testing this condition on γ(t) = e−1/tmone finds that (6) sup 0<t<bm γ(t)γ′′(t) γ′(t)2= sup 0<t<bm1−m+ 1 mtm≤1. We thus have a Tomas-Stein restriction result that includes the surfaces Γ(x) = (x, e−1/|x|m) in R3. The purpose of this paper is to obtain restriction estimates for convex surfaces of revolution in R3. A major role is played by the function γ(t)γ′′(t) γ′(t)2 and our results only require the boundedness of certain Lp0norms of this function. In particular, we obtain a Tomas-Stein restriction estimate for surfaces of revolution in R3satisfying (5). We find it useful to prove our results in a little more general setting. In Section 2 we introduce a family of measures dσγ, state a general (Lp, Lq) restriction result for such measures, and obtain as a corollary the result on Γ(x) = (x, e−1/|x|m). In Section 3 we present the main component of our proof. In Section 4 we prove our results. 2. Statement of results Let 0 < b ≤ ∞, and denote by B(0, b) the ball in R2of center 0 and radius b. Let C([0, b)) be the set of all real-valued functions γ∈C3([0, b)) such that γ(0) = γ′(0) = 0, γ′′(t)>0 for 0 < t < b, and γ(3)(t)≥0 for 0 ≤t < b. Suppose 0 ≤λ≤1, 1 ≤p,p0≤ ∞, 4 ≤q≤∞, and 1/p+2/q ≤1. For γ∈ C([0, b)), let dσγbe the pushforward under the map x→ (x, γ(|x|) of the two-dimensional measure (7) γ′(|x|)3−2λγ′′(|x|)λ |x|γ(|x|)1−λp′ 2q dx with the understanding that when p′=q=∞,p′/(2q) is set to be equal to 1/4; so that p′/(2q) = 1/4 on the sharp line 1/p + 2/q = 1 including the point (1/p, 1/q) = (1,0). Restriction to Convex Surfaces 75 Theorem 1. If 1/p + 2/q = 1 −1/p0, then (8) k[ fdσγkLq(R3)≤Cqγ(|·|)γ′′(|·|) γ′(|·|)2λ 2qLp0(B(0,b)) kfkLp(dσγ) for all (f, γ)∈C0(R3)×C([0, b)), where Cq= 4(27/6π)3/(2q). Notice that if λ= 1, then the density of the measure (7) is |Kγ(|·|)(x)|p′/(2q), so if in addition 1/p + 2/q = 1, then dσγis the same affine surface area measure we described in Section 1. Corollary 1. Suppose γ∈ C([0, b)) is such that γ(|·|)γ′′(|·|) γ′(|·|)21 2qLp0(B(0,b)) <∞. Let λ= 1 and dσ =dσγ. If 1/p + 2/q = 1 −1/p0, then kd fdσkLq(R3).kfkLp(dσ) for all f∈Lp(dσ). For example if γ(t) = e−1/tm, then by (6), γ(|·|)γ′′(|·|) γ′(|·|)21 2qLp0(B(0,bm)) ≤(πb2 m)1/p0<∞ for 1 ≤p0≤ ∞, and so the adjoint restriction estimate in Corollary 1 holds for γ(t) = e−1/tmwhenever 4 ≤q≤ ∞ and 1/p + 2/q ≤1. If, as another example, we take γ(t)=−tlog(1−t), which is in C([0,1)), then γ(|·|)γ′′(|·|) γ′(|·|)21 2qLp0(B(0,1)) ≈−log(1 −|·|) |·| 1 2qLp0(B(0,1)) is finite for 1 ≤p0<∞but not for p0=∞(except if q=∞), and so the adjoint restriction estimate in Corollary 1 holds for γ(t) = −tlog(1 −t) whenever 4 ≤q≤ ∞ and 1/p + 2/q < 1. 3. Main estimate Let ˜ B=B(0, b)∩ {x= (x1, x2)∈R2:x1, x2>0}. The purpose of this section is to prove the following proposition. 76 F. Abi-Khuzam, B. Shayya Proposition 1. Suppose 0< b ≤ ∞ and γ∈ C([0, b)). Then Z˜ BZ˜ B h(u+v, γ(|u|) + γ(|v|)) γ′(|u|)3 |u|γ(|u|) γ′(|v|)3 |v|γ(|v|)1 4 du dv ≤(27/6π)3/2khkL1(R3) for all Lebesgue measurable h:R3→[0,∞]. Proof: Denoting the integral on the left-hand side of the inequality by I, and changing into polar coordinates, we have I=Zb 0Zb 0Zπ 2 0Zπ 2 0 h(reiθ+seiφ, γ(r)+γ(s)) dθ dφr3γ′(r)3s3γ′(s)3 γ(r)γ(s)1 4 dr ds. The change of variable x=reiθ +seiφ (cf [7]) shows that Zπ 2 0Zθ 0 h(reiθ +seiφ, γ(r) + γ(s)) dφ dθ ≤Z√r2+s2<|x|<r+s 2h(x, γ(r) + γ(s)) p(|x|2−(r−s)2)((r+s)2−|x|2)dx. So Zπ 2 0Zπ 2 0 h(reiθ +seiφ, γ(r) + γ(s)) dφ dθ ≤Z√r2+s2<|x|<r+s 4h(x, γ(r) + γ(s)) p(|x|2−(r−s)2)((r+s)2−|x|2)dx ≤Z√r2+s2<|x|<r+s 4h(x, γ(r) + γ(s)) p(2rs)((r+s)2−|x|2)dx ≤Z|x|<r+s 2h(x, γ(r) + γ(s)) (rs)3 4pr+s−|x|dx, Restriction to Convex Surfaces 77 where we have used the inequality r+s≥2√rs. It follows that I≤2Zb 0Zb 0Z|x|<r+s h(x, γ(r) + γ(s)) pr+s−|x|dx γ′(r)3γ′(s)3 γ(r)γ(s)1 4 dr ds = 2 ZB(0,2b)Zb 0Zb 0 h(x, γ(r)+γ(s)) χE(r, s) √r+s−xγ′(r)3γ′(s)3 γ(r)γ(s)1 4 dr ds dx = 4 ZB(0,2b)Zb 0Zb 0 h(x, γ(r)+γ(s)) χF(r, s) √r+s−xγ′(r)3γ′(s)3 γ(r)γ(s)1 4 dr ds dx = 4 ZB(0,2b) II dx, where E={(r, s)∈(0, b)×(0, b) : r+s > |x|},F={(r, s)∈E:s < r}, and II = Zb 0Zb 0 h(x, γ(r) + γ(s)) χF(r, s) √r+s−xγ′(r)3γ′(s)3 γ(r)γ(s)1 4 dr ds. To estimate II, we shall first apply the change of variable r=r(t, y) = γ−1(ysin2t) s=s(t, y) = γ−1(ycos2t), which is defined on the open set Ω = n(t, y)∈R2:π 4< t < π 2, y > 0o; so, with a slight abuse of notation, (r, s) is now a mapping from Ω to R2. The Jacobian of this mapping is J(r,s)(t, y) = 2ysin tcos3t+ 2ysin3tcos t γ′(γ−1(ysin2t))γ′(γ−1(ycos2t)) =ysin 2t γ′(r)γ′(s). But2 γ(r) = ysin2tand γ(s) = ycos2t,(9) so γ′(r)∂r ∂t =ysin 2tand γ′(s)∂s ∂t =−ysin 2t,(10) 2To simplify the notation, we are writing r,s,∂r/∂t, and ∂s/∂t for r(t, y), s(t, y), ∂r/∂t(t, y), and ∂s/∂t(t, y) respectively. 78 F. Abi-Khuzam, B. Shayya and so ysin 2t=pγ′(r)γ′(s)s∂r ∂t  ∂s ∂t . Thus J(r,s)(t, y) = 1 pγ′(r)γ′(s)s∂r ∂t  ∂s ∂t . But also γ′(r)γ′(s) γ(r)γ(s) ∂r ∂t  ∂s ∂t =y2sin22t (ysin2t)(ycos2t)= 4, so γ′(r)3γ′(s)3 γ(r)γ(s)1 4 J(r,s)(t, y) = 4∂r ∂t  ∂s ∂t 1 4 . Next, to determine the domain of integration in the ty-plane, we make the following observations. By the convexity of γ,γ(r) + γ(|x|−r), as a function of r, increases on the interval (|x|/2,|x|). So 2γ(|x| 2)≤γ(r) + γ(|x|−r)< γ(r) + γ(s) whenever |x|/2< r < |x|and |x| − r < s, which are in turn satisfied whenever s < r < |x|< r +s. Also by the convexity of γ, 2γ(|x| 2)≤γ(|x|)≤γ(r)< γ(r) + γ(s) whenever r≥ |x|and s > 0. Thus 2γ(|x| 2)< γ(r) + γ(s)<2γ(b) whenever 0 < s < r < b and |x|< r +s. But, by the definition of the mapping (r, s), y=γ(r) + γ(s) for all (t, y)∈Ω, so 2γ(|x| 2)< y < 2γ(b) whenever 0 < s < r < b and |x|< r +s. For any such (fixed) y, the range of (r, s) is a curve in R2that “enters” the closure of the domain Restriction to Convex Surfaces 79 of integration of II when t=π/4 (i.e. when s=r) and “leaves” when t=τ(y) for some τ(y)∈(π/4, π/2]. Thus II = Z2γ(b) 2γ(|x| 2)Zτ(y) π 4 h(x, y)1 pr+s−|x|4∂r ∂t  ∂s ∂t 1 4 dt dy =Z2γ(b) 2γ(|x| 2) h(x, y)Zτ(y) π 4 √2 pr+s−|x|∂r ∂t  ∂s ∂t 1 4 dt dy. Now, by the definition of τ(y), r+s=r(t, y) + s(t, y)≥ |x|for π 4≤t≤τ(y), so, in particular, r(τ(y), y) + s(τ(y), y)≥ |x|, and hence r+s−|x| ≥ r+s−(r(τ(y), y) + s(τ(y), y)) for π 4< t < τ(y). Thus II≤Z2γ(b) 2γ(|x| 2) h(x, y) Zτ(y) π 4 √2 pr+s−r(τ(y), y)−s(τ(y), y)∂r ∂t  ∂s ∂t 1 4 dt dy. The rest of the proof will be devoted to estimating 1 pr+s−r(τ(y), y)−s(τ(y), y)∂r ∂t  ∂s ∂t 1 4 for 2γ(|x|/2) < y < 2γ(b) and π/4< t < τ(y). We start by examining the function ∂r/∂t +∂s/∂t. By (10), ∂r ∂t +∂s ∂t =ysin 2t γ′(r)−ysin 2t γ′(s) is negative for π/4< t < π/2 (since γ′(s)< γ′(r)), so  ∂r ∂t +∂s ∂t =ysin 2t γ′(s)−ysin 2t γ′(r) = 2ycos t γ′(s)sin t−sin t γ′(r)cos t = 2yscos2t γ′(s)2+sin2t γ′(r)2sin(t−φ),