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41
On the multilinear restriction and Kakeya conjectures
 Acta Math
"... Abstract. We prove dlinear analogues of the classical restriction and Kakeya conjectures in Rd. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the correspondin ..."
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Cited by 52 (12 self)
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Abstract. We prove dlinear analogues of the classical restriction and Kakeya conjectures in Rd. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variablecoefficient problems and the socalled “joints ” problem, as well as presenting some nlinear analogues for n < d. 1.
ON THE ROLE OF QUADRATIC OSCILLATIONS IN NONLINEAR SCHRÖDINGER EQUATIONS II. THE L²critical Case
, 2004
"... We consider a nonlinear semi–classical Schrödinger equation for which quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. The relevance of the nonlinearity was discussed by R. Carles, C. Fermanian–Kammerer and I. Gallagher for L 2supercritical power ..."
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Cited by 42 (6 self)
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We consider a nonlinear semi–classical Schrödinger equation for which quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. The relevance of the nonlinearity was discussed by R. Carles, C. Fermanian–Kammerer and I. Gallagher for L 2supercritical powerlike nonlinearities and more general initial data. The present results concern the L 2critical case, in space dimensions 1 and 2; we describe the set of nonlinearizable data, which is larger, due to the conformal invariance. As an application, we precise a result by F. Merle and L. Vega concerning finite time blow up for the critical Schrödinger equation. The proof relies on linear and nonlinear profile decompositions.
Restriction theory of Selberg’s sieve, with applications, to appear, Journal de Theorie de Nombres de Bordeaux
"... Abstract. The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 –L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime ktuples. Let a1,..., ak and b1,. ..."
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Cited by 35 (10 self)
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Abstract. The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 –L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime ktuples. Let a1,..., ak and b1,...,bk be positive integers. Write h(θ): = ∑ n∈X e(nθ), where X is the set of all n � N such that the numbers a1n + b1,..., akn + bk are all prime. We obtain upper bounds for ‖h ‖ L p (T), p> 2, which are (conditionally on the prime tuple conjecture) of the correct order of magnitude. As a second application we deduce from Chen’s theorem, Roth’s theorem, and a transference principle that there are infinitely many arithmetic progressions p1 < p2 < p3 of primes, such that pi + 2 is either a prime or a product of two primes for each i = 1, 2, 3.
Sharp linear and bilinear restriction estimates for the paraboloid in the cylindrically symmetric case
, 2007
"... In this paper, for cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and linear adjoint restriction estimates. As corollaries of them, for such functions, we first extend the ranges of exponents for the classical linear or bilinear adjoint ..."
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Cited by 20 (2 self)
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In this paper, for cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and linear adjoint restriction estimates. As corollaries of them, for such functions, we first extend the ranges of exponents for the classical linear or bilinear adjoint restriction conjectures, which are sharp up to certain endpoints; Secondly we show that the linear adjoint restriction conjecture for the paraboloid holds for all cylindrically symmetric functions; Lastly, we interpret the restriction estimates in terms of the solutions to the Schrödinger equation. Analogously, we also establish the restriction estimates when the paraboloid is replaced by the lower third of the sphere.
Extension theorems for the Fourier transform associated with nondegenerate quadratic surfaces in vector spaces over finite fields
 Illinois J. Math
"... 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 fiel ..."
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Cited by 12 (5 self)
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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 TomasStein 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.
Extension theorems for paraboloids in the finite field setting
"... In this paper we study the L p − L r boundedness of the extension operators associated with paraboloids in F d q, where Fq is a finite field of q elements. In even dimensions d ≥ 4, we estimate the number of additive quadruples in the subset E of the paraboloids, that is the number of quadruples ( ..."
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Cited by 11 (5 self)
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In this paper we study the L p − L r boundedness of the extension operators associated with paraboloids in F d q, where Fq is a finite field of q elements. In even dimensions d ≥ 4, we estimate the number of additive quadruples in the subset E of the paraboloids, that is the number of quadruples (x, y, z, w) ∈ E 4 with x + y = z + w. As a result, in higher even dimensions, we obtain the sharp range of exponents p for which the extension operator is bounded, independently of q, from L p to L 4 in the case when −1 is a square number in Fq. Using the sharp L p − L 4 result, we improve upon the range of exponents r, for which the L 2 − L r estimate holds, obtained by Mockenhaupt and Tao ([9]) in even dimensions d ≥ 4. In addition, assuming that −1 is not a square number in Fq, we extend their work done in three dimension to specific odd dimensions d ≥ 7. The discrete Fourier analytic machinery
Restriction and spectral multiplier theorems on asymptotically conic manifolds
"... Abstract. The classical SteinTomas restriction theorem is equivalent to the statement that the spectral measure dE(λ) of the square root of the Laplacian on Rn is bounded from Lp(Rn) to Lp (Rn) for 1 ≤ p ≤ 2(n+1)/(n+3), where p′ is the conjugate exponent to p, with operator norm scaling as λn(1/p−1 ..."
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Cited by 11 (5 self)
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Abstract. The classical SteinTomas restriction theorem is equivalent to the statement that the spectral measure dE(λ) of the square root of the Laplacian on Rn is bounded from Lp(Rn) to Lp (Rn) for 1 ≤ p ≤ 2(n+1)/(n+3), where p′ is the conjugate exponent to p, with operator norm scaling as λn(1/p−1/p ′)−1. We prove a geometric generalization in which the Laplacian on Rn is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setting. It is closely related to, but stronger than, Sogge’s discrete L2 restriction theorem, which is an O(λn(1/p−1/p ′)−1) estimate on the Lp → Lp operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including BochnerRiesz summability results, which are sharp for p in the range above. The paper is in three parts. In the first part, we show at an abstract level that restriction estimates imply spectral multiplier estimates, and are implied by certain pointwise bounds on the Schwartz kernel of λderivatives of the spectral measure. In the second part, we prove such pointwise estimates for the spectral measure of the square root of Laplacetype operators on asymptotically conic manifolds. These are valid for all λ> 0 if the asymptotically conic manifold is nontrapping, and for small λ in general. In the third part, we observe that Sogge’s estimate on spectral projections is valid for any complete manifold with C ∞ bounded geometry, and in particular for asymptotically conic manifolds (trapping or not), while by contrast, the operator norm on dE(λ) may blow up exponentially as λ → ∞ when trapping is present. This justifies the statement that the estimate on dE(λ) is strictly stronger than Sogge’s estimate. Contents
NONLINEAR BRASCAMP–LIEB INEQUALITIES AND APPLICATIONS TO HARMONIC ANALYSIS
, 906
"... Abstract. We use the method of inductiononscales to prove certain diffeomorphism invariant nonlinear Brascamp–Lieb inequalities. We provide applications to the multilinear restriction theory for the Fourier transform, and to higher dimensional analogues of a recent convolution inequality of Bejena ..."
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Cited by 6 (3 self)
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Abstract. We use the method of inductiononscales to prove certain diffeomorphism invariant nonlinear Brascamp–Lieb inequalities. We provide applications to the multilinear restriction theory for the Fourier transform, and to higher dimensional analogues of a recent convolution inequality of Bejenaru, Herr and Tataru. Our approach builds on work of Bejenaru et al. and Carbery, Wright and the first author. 1.
Wolff’s inequality for hypersurfaces
 Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial
, 2004
"... We extend Wolff’s “local smoothing ” inequality [18] to a wider class of not necessarily conical hypersurfaces of codimension 1. This class includes surfaces with nonvanishing curvature, as well as certain surfaces with more than one flat direction. An immediate consequence is the L pboundedness of ..."
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Cited by 5 (2 self)
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We extend Wolff’s “local smoothing ” inequality [18] to a wider class of not necessarily conical hypersurfaces of codimension 1. This class includes surfaces with nonvanishing curvature, as well as certain surfaces with more than one flat direction. An immediate consequence is the L pboundedness of the corresponding Fourier multiplier operators. Mathematics Subject Classification: 42B08, 42B15. The purpose of this article is to extend the “local smoothing ” inequality of [18], [5], proved there for circular cones, to a wider class of bounded surfaces of codimension 1 in R d+1, d ≥ 2. Recall that Wolff’s inequality [18] states that if f is a function with Fourier transform supported in a δneighbourhood of the segment of the circular cone with 1 ≤ x  ≤ 2, then ∀ǫ ∃Cǫ: ‖f‖p ≤ Cǫδ d−1 d − + 2 p −ǫ ‖f‖p,δ, (1) with d = 2 and p> 74; this was then extended in [5] to d ≥ 3, p> 2 + 32 3d−7, and d ≥ 4, p> 2 + 8. Here the norm on the right side is defined by d−3