SOME CONVOLUTION INEQUALITIES AND THEIR APPLICATIONS

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 6, Pages S )91-X Article electronically published on November 3, 1 SOME CONVOLUTION INEQUALITIES AND THEIR APPLICATIONS DANIEL M. OBERLIN Abstract. We introduce a class of convolution inequalities and study the implications of these inequalities for certain problems in harmonic analysis. 1. Introduction Let λ be a nonnegative Borel measure on R n. Two familiar problems in harmonic analysis are C) determine the indices p and q such that the convolution estimate λ f q Cλ, p, q) f p holds for f L p R n ), and R*) determine the indices p and q such that the adjoint Fourier restriction estimate fdλ q Cλ, p, q) f L p λ) holds for f L p R n ). Results on problems C) and R*) generally deal either with pushing the range of indices for which the estimates hold in cases where the measures are nondegenerate or with extending such previously-known results to cover degenerate cases. The results in this paper are of the latter type. In particular, our interest here is to study the implications for problems C) and R*) of certain inequalities which λ might satisfy. Our study of these inequalities was originally motivated by the hope that, with the proper choice of measure λ on the possibly degenerate manifold M, there would exist convolution and restriction estimates which are uniform over large classes of M. And this hope was, in turn, motivated by the paper [8] of Drury, where the suitability to problems in harmonic analysis of affinely invariant measures was first pointed out. To begin, we will say that λ satisfies 1) if, letting E denote the Lebesgue measure of E R n in this paper E will always denote a Borel subset of R n ), we have 1) λ χ E L λ) c E 1 for some positive constant c and all E R n. We will also be interested in a weakened version of 1): if there exists some real-valued Borel function ω on suppλ) Received by the editors May, 1 and, in revised form, June 1, 1. Mathematics Subject Classification. Primary 4B1. Key words and phrases. Convolution, restriction. The author was partially supported by the NSF. 541 c 1 American Mathematical Society

2 54 DANIEL M. OBERLIN such that the inequality ) χ E y y 1 )dλy 1 )) dλy ) c E {ωy 1) ωy )} holds for some positive constant c and all E R n, then we will say that λ satisfies ). We will prove the following theorems. Theorem 1. If λ satisfies 1), then the inequality λ f 3 Cc) f 3 f L 3 R n ). holds for Theorem. If λ satisfies ), then the inequality λ χ E 3 Cc) E 3 E R n. holds for Theorem 3. If λ satisfies 1), then the inequality χ A dλ) χ B dλ) L, R n ) cλa) 1 4 λb) 1 4 holds for Borel subsets A and B of the support of λ. Theorem 4. If λ satisfies ), then the inequality fdλ q Cc, p) f L p λ) holds for f L p λ) whenever 1 p<4 and 1 p + 3 q =1. Notes. a) The estimate in Theorem is equivalent to a weak-type version of the estimate in Theorem 1. b) We will prove Theorem 4 by observing that ) implies the conclusion of Theorem 3 when A = B which inequality we regard as a weak endpoint estimate for the adjoint Fourier restriction problem) and then showing how that inequality implies the conclusion of Theorem 4. c) Concerning the organization of this paper, the proofs for the theorems will be found in. Proofs for the examples are in 3. Here is an easy example of a measure λ satisfying 1): let dλ be dt on the curve t, t )inr. Then, for E R, 1 dt χ E t s, t s ) ds) χ E t s, t s ) t s dsdt = E, giving 1) with c = 1/. Of course the convolution and restriction results which follow for this λ from Theorems 1-4 are, except perhaps for Theorem 3, not new. But our study of inequalities like 1) and ) began with a desire to obtain similar results with constants uniform over a large class of measures on curves generalizing dt on t, t ). Indeed, the proofs for Theorems and 4 are simple abstractions of parts of the proofs in [16] and [19], respectively see the discussion after Example 1 below), while the proof of Theorem 1 bears the same relationship to the proof of Theorem 1 in [15]. Here are five examples of measures satisfying 1) or ): Example 1. Suppose φ is a real-valued C ) function on a, b) withφ positive and nondecreasing. Let dλ be the measure φ t) 1/3 dt on the curve t, φt) ) in R. Then, for E R,wehave b b ) φ 3) χ E t s, φt) φs) φ s) ds) 1/3 t) 1/3 dt 4 E. a t

3 SOME CONVOLUTION INEQUALITIES AND THEIR APPLICATIONS 543 Here the measure λ is called the affine arclength measure on the curve t, φt) ). As previously mentioned, its study in harmonic analysis was initiated by Drury [8]. Thus this example shows that ) holds uniformly for affine arclength measure on a large class of curves in R with ωt) =t). The uniform convolution and restriction results furnished by this example and Theorems and 4 are the main results in [16] and [19], respectively, while the proof of 3) can be found in either of those papers. It can be shown that there are examples of this type for which the stronger inequality 1) fails. It is an open question whether a uniform version of Theorem 1 holds anyway. Example. If p is a real-valued polynomial of degree N and p is of constant sign on an interval I, then 1) holds uniformly for affine arclength measure on the graph of p over I: ) ) p χ E t s, pt) ps) p s) 1 3 ds t) 1 3 dt CN) E. I I Generalizing the t, t ) example, this is a consequence of Lemma 3.1 in [] and an argument originally appearing in [15]. The uniform convolution estimate which then follows from Theorem 1 was originally proved in [18]]. The following definition is required for Examples 3, 6, and 8: Definition. Suppose Ω R p. Say that a mapping F : Ω R p has generic multiplicity l if card F 1 y) l for almost all y R p. Example 3. Suppose D is an open subset of C and that φ : D C is analytic. Suppose that φ andthemapz,w) z w, φz) φw) ) from D into C both have generic multiplicities l. Letdw be two-dimensional Lebesgue measure on C. Then the following inequality holds for E C with constant c depending only on l: ) φ χ E z w, φz) φw) φ w) dw) /3 z) /3 dz c E. D {w D: φ w) φ z) } Thus ) holds with dλ = φ z) /3 dz on the graph { z,φz) ) : z D} and with ωz) = φ z). This is an analogue of Example 1. Earlier convolution and restriction estimates for measures on -dimensional manifolds in R 4 can be found in [9] and [5] respectively. There are also examples where 1) holds in this setting: Example 4. Suppose that D is an open and convex subset of C. Suppose additionally that φ is a polynomial of degree N and that there is a constant A such that the inequality 4) w z 1 φ z + tw z) ) dt A φ z) φ w) holds for z,w D. ThenforE C there is the inequality ) φ χ E z w, φz) φw) φ w) dw) /3 z) /3 dz c E D D with c = ca, N).

4 544 DANIEL M. OBERLIN For example, if φz) =z N and D is a sector with vertex and angle π 4N,then 4) holds. Convolution and restriction results such as Theorems 1 4 are most familiar when, as in Examples 1 4, n is even and λ is a measure on an n -dimensional manifold in R n. Here is an example on a degenerate) -manifold in R 3, the light cone. Example 5. Consider the cone C = {y, y ) :y R } R +1 = R 3. Then the inequality ) y 5) χ E y y 1, y y 1 ) y 1 1 dy 1 1 dy c E { y 1 y } holds for E R 3. This is ) with dλ = y 1 dy on C and with ωy, y ) = y 1. The convolution result implied by Example 5 and Theorem is weaker than the appropriate special case of the main theorem in [1] see also Theorem 1 in [14]). The restriction result which follows from Theorem 4 is originally due to Taberner [3], though the current proof is simpler than his or the proof in [1]. An analogue of 5) also holds for the measure dsdt on the surface s t, s t,s 3 t 3 )inr 3. Theorems 1 4 admit certain generalizations. For example, the two following results are extensions of Theorems 1 and, respectively. Theorem 5. Suppose that λ is a nonnegative Borel measure on R n satisfying the inequality 6) λ χ E L m λ) c E m 1 m for E R n. Then the convolution estimate λ f m+1 Cc) f m+1 m holds for f L m+1 m R n ). Theorem 6. Suppose that λ is a nonnegative Borel measure on R n. Suppose that ω is a real-valued Borel function on suppλ) and suppose the inequality ) mdλy 7) χ E y y 1 )dλy 1 ) ) c E m 1 {ωy 1) ωy )} holds for E R n. Then the convolution estimate λ χ E m+1 Cc) E m m+1 holds whenever E R n. When m = n, Theorems 5 and 6 describe convolution estimates which one might expect when λ is a measure on an n 1)-dimensional manifold in R n. Here are two examples: Example 6. Suppose φ is a real-valued C ) function defined on a subset Ω of R n 1. Define γ :Ω R n by γy) = y, φy) ). Then the inequality ) 1 n 1 χ E y y 1 ) det H φ y 1 ) n+1 dy1 det Hφ y ) n+1 dy Ω { det H φ y 1) det H φ y ) } c E n 1

5 SOME CONVOLUTION INEQUALITIES AND THEIR APPLICATIONS 545 holds for E R n with constant c depending only on n and the generic multiplicities of the gradient of φ and the map Ψ:Ω n R n ) n 1, Ψy 1,...,y n )= γy 1 ) γy ),...,γy 1 ) γy n ) ). Thus 7) holds with m = n and ωx) = det H φ x). This is ) in [17]. The conclusion that follows from Example 6 and Theorem 6 is the principal result of [17].) The stronger hypothesis 6) of Theorem 5 seems substantially more difficult to verify in any generality. The next example may be compared with Example 5: Example 7. Consider the paraboloid P = {y, y ):y R } R +1 = R 3.Then the inequality ) 3dy χ E y y 1, y y 1 )dy 1 c E holds for E R 3. If dλ = dx on P,thisis6)withm = 3. The convolution result which follows from Example 7 and Theorem 5 is well-known. But, as will be pointed out after the proof of Theorem 5, the hypothesis of Theorem 5 actually yields a slightly stronger conclusion. Not surprisingly, the inequalities of Examples 6 and 7 seem irrelevant to Fourier restriction. But, following a suggestion of A. Seeger and S. Wainger, we will prove the following result: Theorem 7. Suppose that λ is a nonnegative Borel measure on R n satisfying the inequality ) m+ 8) χ E y 1 + y + + y m )dλy ) dλy m ) dλy 1 ) c E {ωy j) ωy 1)} for some nonnegative integer m 3, some real-valued Borel function ω on suppλ), and all E R n. Then the adjoint restriction estimate 9) fdλ q Cc, p) f L p λ) holds whenever 1 p + mm+1) 1 q =1and 1 p<m+. Example 8. Let γt) =γ 1 t),...,γ n t)) : [a, b] R n parametrize a C n) curve in R n.writedt) = det[γ t),...,γ n) t)]. Let t 1,...,t n )= det[γ t 1 ),...,γ t n )] and let V t 1,...,t n ) be the absolute value of) the Vandermonde determinant given by t 1,...,t n )whenγ j t) =t j. Suppose there is a positive constant A such that A 1 t 1,...,t n ) V t 1,...,t n ) sup Dt j ) A 1 t 1,...,t n ) 1 j n for t j [a, b]. Then there is a constant c, depending only on A, n, and the generic multiplicity of the map t 1,...,t n ) γt 1 )+ + γt n ), such that the inequality χ E γt1 )+ + γt n ) ) n ) n+ Dt j ) nn+1) dt dt n {Dt j) Dt 1)} holds for E R n. Dt 1 ) nn+1) dt1 c E

6 546 DANIEL M. OBERLIN That is, if λ is the measure on the curve γ given by dλ = Dt) nn+1) dt, thenλ satisfies 8) with m = n and ω γt) ) = Dt) 1.AsinExample1,themeasureλis affine arclength on γ. The range of p in the adjoint restriction estimate furnished by Theorem 8 is short of the conjectured optimal range see Drury [6]) and also short of the range in Drury s three-dimensional result for degenerate curves Theorem 3 in [8]) but the same as the range in Christ s local result A) of Theorem 1.1 in [5]) for nondegenerate curves. The hypothesis 8) is particularly easy to verify when [a, b] [, ), γ j t) =t j for 1 j<n,andγ n t) is a polynomial with nonnegative coefficients.. Proofs of the Theorems Proof of Theorems 5 and 1. The proof is an adaptation of an idea from the proof of Theorem 1 in [15]. See [1] and [] for other applications of the same idea. Assuming the hypothesis 6), we will actually establish the multilinear estimate m+1 i=1 λ f i x)dx Cc) m+1 i=1 f i m+1,m+1, m where the norms on the right hand side of the inequality are Lorentz-space norms, the functions f i are nonnegative and Borel measurable, and dx denotes Lebesgue measure on R n. By symmetry and a multilinear argument of Christ [4] see [7] for a more detailed exposition of Christ s idea), it is sufficient to establish the estimate 1) m+1 i=1 m+1 λ f i x)dx Cc) f 1 1 i= f i m m 1,1. We may assume that, for i m +1, each f i is of the form χ Ei for Borel E i R n. An application of Fubini s theorem to the left hand side of the inequality above leads to m+1 f 1 x) ) m+1 λ χei x)y) dλy)dx f 1 1 λ χ Ei x) L m dλ). i= According to 6) this is bounded by m+1 Cc) f 1 1 i= E i m 1 m, which is the right hand side of 1). This proves Theorem 5. i= The proof shows that the Lebesgue space norm m+1 m Theorem 5 may be replaced by the weaker Lorentz space norm m+1 m in the conclusion of,m+1. That is the slightly stronger conclusion mentioned after Example 7. Of course Theorem 1 is the special case m =oftheorem5.

7 SOME CONVOLUTION INEQUALITIES AND THEIR APPLICATIONS 547 Proof of Theorems 6 and. We estimate m+1dx λ χ E m+1 m+1 = χ E x y)dλy)) m +1) χ E x y ) χ E x y 1 )dλy 1 )) mdλy ) dx =m +1) χ E x) {ωy 1) ωy )} χ E x + y y 1 )dλy 1 )) mdλy ) dx m c E m {ωy 1) ωy )} by the hypothesis 7) and the fact that E x = E. This argument is just an abstraction of the first part of the proof of Theorem 1 in [17]. Again, Theorem is the special case m =oftheorem6. Proof of Theorem 4. By interpolation with the case p = 1 it is enough to establish the estimate χ A dλ q, Cc, p) λa) 1 p 11) for all Borel subsets A of suppλ) whenever 1 p<4and 1 p + 3 q =1. ByHunt s generalization of the Hausdorff-Young theorem, this will follow from the inequality χ A dλ) χ A dλ) r, Cc, p) λa) p 1) if 1 p = 3 1 r 1 so that 1 r + q = 1). Now 1) is true for all p, r) of interest if it is true for the two extreme cases p, r) =1, 1) and p, r) =4, ). The first of these is easy, and so it is enough to establish the estimate 13) χ A dλ) χ A dλ), Cc) λa) 1, which we regard as a weak endpoint version of 11). This inequality will follow from 14) χ E y y 1 )dλy 1 )dλy ) Cc) λa) 1 E 1, A A for E R n. But the the left hand side of 14) is bounded by χ E y y 1 )dλy 1 )dλy ) A {ωy 1) ωy )} + A {ωy 1) ωy )} χ E) y y 1 )dλy 1 )dλy ). Applying Hölder s inequality and then ) to each of these in turn yields 14). Again, this argument is just an abstraction of part of the argument in [19]. Now, as noted in b) after the statement of Theorem 4, 13) and therefore the conclusion of Theorem 4 clearly follow from the conclusion of Theorem 3. Proof of Theorem 3. It is enough to check that χ E y y 1 )dλy 1 )dλy ) Cc) λa) 1 4 λb) 1 4 E 1 A B

8 548 DANIEL M. OBERLIN for E R n.but,byhölder s inequality and 1), the left hand side is bounded by and, similarly, by cλa) 1 E 1. λb) 1 λ χ E) L λ) cλb) 1 E 1 Proof of Theorem 7. This is a generalization of the proof of Theorem 4. By interpolation with the case p = 1 it is enough to establish an endpoint analogue of 9): 1 m+ χ A dλ mm+) Cc) λa), for Borel subsets A of suppλ). By Hunt s generalization of the Hausdorff-Young theorem, this will follow from the inequality χ A dλ) χ A dλ) 1 1 m m+ m+ Cc) λa), m, where the convolution on the left hand side is m fold. This estimate follows from χ E y y m )dλy 1 ) dλy m ) m A A A {ωy j) ωy 1)} Cc)λA) m m+ E wherewehaveusedhölder s inequality and 8). ) χ E y y m )dλy ) dλy m ) dλy 1 ) m+, 3. Proofs for the Examples Proof for Example. Let T be the mapping from the Lorentz space L,1 R )into L I, p t) 1 3 dt) defined by Tft) = f t s, pt) ps) ) p s) 1 3 ds. I We regard the desired inequality ) ) p χ E t s, pt) ps) p s) 1 3 ds t) 1 3 dt CN) E I I as an estimate for T, and we will prove it by establishing the dual estimate 15) T g L, R ) CN) g L I, p t) 1. 3 dt) Because of the convexity of p on I, the mapping t, s) t s, pt) ps) ) from I into R is one-to-one. The absolute value of its Jacobian is p t) p s). Thus the equation f,t g = Tf,g = I I f t s, pt) ps) ) p t)p s) 1/3 p t) p s) gt) p t) p s) ds dt shows that, for y>, the measure of the set { T g >y} is E y p t) p s) ds dt,

9 SOME CONVOLUTION INEQUALITIES AND THEIR APPLICATIONS 549 where E y = {t, s) I : gt) p t)p s) 1/3 p t) p s) }. y Lemma 1 below Lemma 3.1 from []) bounds the inner integral by CN) gt) p t) 1/3 y. Thus an integration in t completes the proof of 15). Lemma 1. There is a constant CN) such that the following is true: If r is a real-valued polynomial of degree N 1 and I is an interval with r of constant sign on I, then for all t I and all B> the inequality rt) rs) ds CN) B r t) 1/3 holds. {s I:B r t)r s) 1/3 rt) rs) } Proof for Example 3. With notation as in Example 3, we need to establish the inequality D {w D: φ w) φ z) } χ E z w, φz) φw) ) φ w) /3 dw) φ z) /3 dz c E for E C and with c = cl). The mapping z,w) z w, φz) φw) ) from D D into C has Jacobian φ z) φ w),so ) χ E z w, φz) φw) φ z) φ w) dw dz cl) E. D D Thus it is enough to show that φ z) 3 {w D: φ w) φ z)} D or that φ z) 3 D χ E z w, φz) φw) ) φ w) 3 dw ) ) cl) χ E z w, φz) φw) φ z) φ w) dw {w D: φ w) φ z)} ) ) χ E z w, φz) φw) φ w) 4 3 φ w) dw ) cl) χ E z w, φz) φw) φ z) φ w) φ w) φ w) dw. If we write Z = φ z) andw = φ w), so that dw = φ w) dw, andthenlet gw )= w:φ w)=w χ E z w, φz) φw) ) χ{w D: φ w) φ z)}w) φ w) 4 3,

10 55 DANIEL M. OBERLIN so that g 1/ l 1/ φ z) /3, this will follow from the inequality ) gw )dw c g 1/ gw ) 3 W Z dw C for fixed Z C and for nonnegative Borel functions g on C. To see the latter inequality, first note that the function W is in the Lorentz space L, W dw ). Then gw ) dw = gw ) W W dw c g L,1 W dw ) C C C c g 1 4 g 3 4 L 3 W dw ). Proof for Example 4. The proof is analogous to the proof for Example. There are two points of difference: first, the mapping z,w) z w, φz) φw) ) may not be one-to-one. But, by an argument involving Bezout s theorem, it will have generic multiplicity CN). Second, reflecting the fact that the mapping above has Jacobian φ z) φ w), Lemma 1 must be replaced by Lemma. Under the hypothesis 4) there is a constant c = ca, N) such that for all z D and all B>, theinequality 16) φ z) φ w) dw CN) B φ z) /3 holds. {w D:B φ z)φ w) /3 φ z) φ w) } Proof of Lemma. Without loss of generality assume z =. Usingtheconvexity of D, there is a function rθ) with the property 17) φ ) φ rθ)e iθ ) B φ ) φ rθ)e iθ ) 3 for θ<π such that the left hand side of 16) is bounded by 18) Now 19) rθ) π rθ) φ ) φ re iθ ) rdr φ ) φ re iθ ) rdrdθ. rθ) rθ) r rdr φ se ) ds) iθ rθ). φ se ) ds) iθ It follows from the equivalence of norms on the finite-dimensional space of polynomials of degree N that there is CN) such that rθ) φ ) α φ rθ)e iθ ) 1 α CN) rθ) φ se iθ ) ds

11 SOME CONVOLUTION INEQUALITIES AND THEIR APPLICATIONS 551 for α 1. Thus rθ) rθ) ) ) rθ) 4 φ se iθ ) ds φ se iθ ) ds CN) φ ) α φ rθ)e iθ ) α CN) 4 A4 B φ ) φ rθ)e iθ ) 4 3 φ ) α φ rθ)e iθ ) α, where we have used 4) and 17) to obtain the last inequality. Now take α = 1 3 and integrate in θ. With 19) this yields the desired bound for 18). Proof for Example 5. It is notationally convenient to regard R +1 as C R and use coordinates re iθ,t). It will be enough to establish the inequality ) π r π c E 1 χ E re iθ se iφ,r s) dφ ds gr, θ) dθ dr π ) 1 gr, θ) dθ dr for nonnegative Borel functions g. The proof is an easy consequence of two lemmas. Lemma 3. The following inequality holds for some positive C and all nonnegative Borel functions f on, ): 1 r fr s) ds ) dr C fr) dr. r 1/ s 1/ This fractional integration result is 9.9.5) in [13]. Lemma 4. There is a positive constant C such that the following inequality holds for all Borel S R and all r, s, ): π π ) dθ χ S re iθ se iφ C ) dφ rs S. Proof of Lemma 4. Using the idea from the t, t ) example at the beginning of the paper, we have for fixed θ π ) π 1) χ S re iθ se iφ ) dφ C χ S re iθ se iφ ) sinθ φ) dφ. Since a Jacobian computation shows that rs π π χ S re iθ se iφ ) sinθ φ) dφ dθ C S, integrating 1) with respect to θ completes the proof of the lemma. If E R +1 and t R, thene t will stand for {x R :x, t) E}. Since π π χ E re iθ se iφ,r s) dφ gr, θ) dθ π π χ E re iθ se iφ,r s) dφ ) ) 1 dθ π gr, θ) dθ ) 1,

12 55 DANIEL M. OBERLIN Lemma 4 yields the following bound for the left hand side of ): 1 r ds π C E r s 1 gr, θ) dθ ) 1 dr 1 C r 1 r 1 r s 1 ds ) ) E r s 1 1 dr s 1 π gr, θ) dθ dr An application of Lemma 3 now yields ). Proof for Example 7. With standing for, the inequality to be proved is 3 ) χ E s u j,t v j,s + t u j vj ) ds dv 3 c E 1 for E R 3. The proof requires two lemmas. Lemma 5. The following inequality holds for a 1,a,b 1,b R and for Borel S R : χ S u 1 u,u 1 + a 1u 1 + b 1 u a u b 3 )du ) du1 1 S. This is proved by changing variables in the t, t ) example at the beginning of the paper. Lemma 6. The following inequality holds for all nonnegative Borel functions f on, ): v1 v fv 1 )fv ) 1 fv3 ) 1 dv ) 1. dv3 dv v 1 v v 1 1 fv) dv Proof of Lemma 6. We have v v ) fv 3 ) 1 dv3 1 v 1 1 fv 3 ) dv 3, v v and so v1 v fv 3 ) 1 dv3 fv ) 1 dv fv 1 ) dv 1 v 1 v v 1 v1 v ) 1 1 fv 3 ) dv 3 v 1 fv ) 1 dv fv 1 ) dv 1 v 1 v v 1 v1 v 1 ) 1 v 1 fv ) 1 1 dv fv 3 ) dv 3 fv 1 ) dv 1 v 1 v 1 v 1 v1 ) 1 1 v1 ) v dv fv 3 )dv 3 fv 1 ) dv 1 v 1 v 1 v 1, 1 fv) dv) as desired. Let I represent the integral on the left hand side of ) and let I represent that integral restricted to the set { v 3 t v t v 1 t }. ) 1.

13 SOME CONVOLUTION INEQUALITIES AND THEIR APPLICATIONS 553 Then I 3I.InI replace s by s + u 1, t by t + v 1,andthen,forj =1,, 3, replace v j by v j s t u 1. The result is χ E s, t, s + t +tv 1 ) χ E s + u1 u,t+ v 1 v, s + u 1 ) u +t + v 1 s t u 1) v s t u 1) ) χ E s + u1 u 3,t+ v 1 v 3, s + u 1 ) u 3 +t + v 1 s t u 1) v 3 s t u 1) ) ds dv 3, where the integral is restricted to the set { v 3 v 1 t v v 1 t t }. If E R 3 and t R, then we will write E t for {x 1,x 3 ) R :x 1,t,x 3 ) E}. Hölder s inequality and two applications of Lemma 5 bound χ E s + u1 u,t+ v 1 v, s + u 1 ) u +t + v 1 s t u 1) v s t u 1) ) χ E s + u1 u 3,t+ v 1 v 3, s + u 1 ) u 3 +t + v 1 s t u 1) v 3 s t u 1) ) by du 1 du du 3 Thus I is bounded by 1 Et+v1 v 1 Et+v1 v χ E s, t, s + t +tv 1 ) E t+v1 v 1 Et+v1 v 3 1 ds dt dv3 dv dv 1. Replacing t by t v 1 and then v j by v j + t leads to 1 χ E s, v1,s + t v 1 + t) ) E v 1 E v3 1 ds dt dv3 dv dv 1, where indicates that the variables vj are restricted to { v 3 v v 1 }. The change of variables s, t) s, s + t v 1 + t) ) has Jacobian v 1,sothis last integral is bounded by v1 v dv 1 E v1 E v 1 E v3 1 1 dv3 dv v 1 v v 1. Now Lemma 6 completes the proof. Proof for Example 8. The proof is analogous to the one for Example. It depends on the following result. Lemma 7. There is C = Cn) such that the following inequality holds for y>: V t 1,...,t n ) dt dt n Cy n+ n. {V t 1,...,t n)<y}

14 554 DANIEL M. OBERLIN This follows from a homogeneity argument and the observation see the proof of Lemma 1 in [11]) that V t 1,...,t n ) dt dt n <. by {V t 1,...,t n)<1} Let T be the mapping from L n+,1 R n )intol n+ [a, b],dt) /nn+1)) dt) defined Tft 1 )= {Dt j) Dt 1)} where the integral is over f γt 1 )+ + γt n ) ) n Dt j ) nn+1) dt dt n, {t,...,t n ) [a, b] n 1 : Dt ) Dt),...,Dt n ) Dt 1 )}. As in the proof for Example, we regard the conclusion of Example 8 as an estimate for T and establish the dual estimate 3) The equation T g L n+ n, R n ) c g L n+ n [a,b],dt) /nn+1)) dt). f,t g = Tf,g = {Dt j) Dt 1)} f γt 1 )+ + γt n ) ) n Dt j ) nn+1) 1 gt 1 ) t 1,...,t n ) t 1,...,t n ) dt 1 dt n shows that, for y>, the measure of { T g >y} is bounded by c t 1,...,t n ) dt 1 dt n, E y where c depends on n and the generic multiplicity of t 1,...,t n ) γt 1 )+ + γt n ) ) and where E y = {t 1,...,t n ) [a, b] n : gt 1) n Dt j ) nn+1) t1,...,t n ), y 1 Dt j ) Dt 1 )}. Thus 3) will follow from b gt1 ) ) n+ n 4) t 1,...,t n ) dt 1 dt n c Dt 1 ) nn+1) dt1, E y a y where c depends only on n and A. Fort 1 [a, b] andb>letf t1,b be the set n {t,...,t n ) [a, b] n 1 : B Dt j ) nn+1) t1,...,t n ), Dt j ) Dt 1 )}. 1

15 SOME CONVOLUTION INEQUALITIES AND THEIR APPLICATIONS 555 Then 4) will be a consequence of the inequality 5) t 1,...,t n ) dt dt n cb n+ n Dt1 ) F t1,b where c depends only on n and A. Recalling the hypothesis nn+1), A 1 t 1,...,t n ) V t 1,...,t n ) sup Dt j ) A 1 t 1,...,t n ) 1 j n of Example 8, it follows from n 1 Dt j) /nn+1)) Dt 1 ) 1/n+1) that t 1,...,t n ) dt dt n F t1,b A Dt 1 ) Now 5) follows from Lemma 7. {V t 1,...,t n) BA 1 Dt 1) n+1 1 } V t,...,t n ) dt dt n. References 1. J.-G. Bak, An L p -L q estimate for Radon transforms associated to polynomials, Duke Math. J. 11 ), MR 1b:41. J.-G. Bak, D.M. Oberlin, and A. Seeger, Two endpoint bounds for generalized Radon transforms in the plane, Revista Math. to appear). 3. B. B. Taberner, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc ), MR 86k:43 4. M. Christ, Estimates for the k plane transform, Indiana Univ. Math. J ), MR 86k:444 5., On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc ), MR 87b: S.W. Drury, Restrictions of Fourier transforms to curves, Ann. Inst. Fourier, Grenoble ), MR 86e:46 7., A survey of k plane transforms, Contemp. Math ), MR 9b:44 8., Degenerate curves and harmonic analysis, Math. Proc. Camb. Phil. Soc ), MR 91h:41 9. S.W. Drury and K. Guo, Convolution estimates related to surfaces of half the ambient dimension, Math. Proc. Camb. Phil. Soc ), MR 9j:41 1., Some remarks on the restriction of the Fourier transform to surfaces, Math. Proc. Camb. Phil. Soc ), MR 94f:4 11. S.W. Drury and B.P. Marshall, Fourier restriction theorems for curves with affine and Euclidean arclengths, Math. Proc. Camb. Phil. Soc ), MR 87b: J. Harmse, On Lebesgue space estimates for the wave equation, Indiana Univ. Math. J ), MR 91j: G.H. Hardy, J.E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, MR 13:77e 195 ed.) 14. D.M. Oberlin, Convolution estimates for some distributions with singularities on the light cone, Duke Math. J ), MR 91f: , Multilinear proofs for two theorems on circular averages, Colloq. Math ), MR 93m:45 16., Convolution with affine arclength measures in the plane, Proc. Amer. Math. Soc ), MR c: , Convolution with measure on hypersurfaces, Math. Proc. Camb. Phil. Soc. 19 ), MR 1j:414

16 556 DANIEL M. OBERLIN 18., Convolution with measures on polynomial curves, Math. Scand. to appear). 19., Fourier restriction for affine arclengthmeasures in the plane, Proc. Amer. Math. Soc. 19 1), CMP 1:16 Department of Mathematics, Florida State University, Tallahassee, Florida address: