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Journal articles on the topic 'Théorèmes de restriction de Fourier'

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Published: 25 January 2025

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1

Kovač, Vjekoslav. "Fourier restriction implies maximal and variational Fourier restriction." Journal of Functional Analysis 277, no. 10 (November 2019): 3355–72. http://dx.doi.org/10.1016/j.jfa.2019.03.015.

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2

Demeter, Ciprian, and S. Zubin Gautam. "Bilinear Fourier Restriction Theorems." Journal of Fourier Analysis and Applications 18, no. 6 (June 6, 2012): 1265–90. http://dx.doi.org/10.1007/s00041-012-9230-9.

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3

Demeter, Ciprian. "Bourgain’s work in Fourier restriction." Bulletin of the American Mathematical Society 58, no. 2 (January 27, 2021): 191–204. http://dx.doi.org/10.1090/bull/1717.

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4

Kovač, Vjekoslav, and Diogo Oliveira e Silva. "A variational restriction theorem." Archiv der Mathematik 117, no. 1 (May 7, 2021): 65–78. http://dx.doi.org/10.1007/s00013-021-01604-1.

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AbstractWe establish variational estimates related to the problem of restricting the Fourier transform of a three-dimensional function to the two-dimensional Euclidean sphere. At the same time, we give a short survey of the recent field of maximal Fourier restriction theory.

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5

Shayya, Bassam. "Fourier restriction in low fractal dimensions." Proceedings of the Edinburgh Mathematical Society 64, no. 2 (April 30, 2021): 373–407. http://dx.doi.org/10.1017/s0013091521000201.

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AbstractLet $S \subset \mathbb {R}^{n}$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\mathbb {R}^{n}$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \lesssim R^{\alpha }$ for all balls $B_R$ in $\mathbb {R}^{n}$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^{q}$ against the measure $\chi _X \,{\textrm {d}}x$. Our approach consists of replacing the characteristic function $\chi _X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on $\mathbb {R}^{n}$ and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du–Zhang theorem in the range $0 < \alpha < n/2$.

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6

Drury, S. W., and B. P. Marshall. "Fourier restriction theorems for degenerate curves." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 3 (May 1987): 541–53. http://dx.doi.org/10.1017/s0305004100066901.

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Fourier restriction theorems contain estimates of the formwhere σ is a measure on a smooth manifold M in ∝n. This paper is a continuation of [5], which considered this problem for certain degenerate curves in ∝n. Here estimates are obtained for all curves with degeneracies of finite order. References to previous work on this problem may be found in [5].

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7

Bruce, Benjamin Baker. "Fourier restriction to a hyperbolic cone." Journal of Functional Analysis 279, no. 3 (August 2020): 108554. http://dx.doi.org/10.1016/j.jfa.2020.108554.

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8

Carneiro, Emanuel, Diogo Oliveira e Silva, and Mateus Sousa. "Extremizers for Fourier restriction on hyperboloids." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 36, no. 2 (March 2019): 389–415. http://dx.doi.org/10.1016/j.anihpc.2018.06.001.

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9

Nicola, Fabio. "Slicing surfaces and the Fourier restriction conjecture." Proceedings of the Edinburgh Mathematical Society 52, no. 2 (May 28, 2009): 515–27. http://dx.doi.org/10.1017/s0013091507000995.

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AbstractWe deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid and the elliptic hyperboloid in ℝn implies that for the cone in ℝn+1. We also prove a new restriction estimate for any surface in ℝ3 locally isometric to the plane and of finite type.

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10

Carbery, Anthony. "Restriction implies Bochner–Riesz for paraboloids." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 3 (May 1992): 525–29. http://dx.doi.org/10.1017/s0305004100075599.

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Let Σ ⊆ ℝn be a (compact) hypersurface with non-vanishing Gaussian curvature, with suitable parameterizations, also called Σ: U → ℝn (U open patches in ℝn−1). The restriction problem for Σ is the question of the a priori estimate (for f ∈ S(ℝ))(^denoting the Fourier transform). The Bochner-Riesz problem for Σ is the question of whether the functionsdefine Lp-bounded Fourier multiplier operators on ℝn in the range.

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11

Hu, Yi, and Xiaochun Li. "Discrete Fourier restriction associated with KdV equations." Analysis & PDE 6, no. 4 (August 21, 2013): 859–92. http://dx.doi.org/10.2140/apde.2013.6.859.

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12

Mockenhaupt, Gerd. "A restriction theorem for the Fourier transform." Bulletin of the American Mathematical Society 25, no. 1 (July 1, 1991): 31–37. http://dx.doi.org/10.1090/s0273-0979-1991-16018-0.

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13

Magyar, Ákos. "On Fourier restriction and the Newton polygon." Proceedings of the American Mathematical Society 137, no. 02 (August 26, 2008): 615–25. http://dx.doi.org/10.1090/s0002-9939-08-09510-5.

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14

Stovall, Betsy. "Extremizability of Fourier restriction to the paraboloid." Advances in Mathematics 360 (January 2020): 106898. http://dx.doi.org/10.1016/j.aim.2019.106898.

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15

Oliveira e Silva, Diogo. "Extremizers for Fourier restriction inequalities: Convex arcs." Journal d'Analyse Mathématique 124, no. 1 (October 2014): 337–85. http://dx.doi.org/10.1007/s11854-014-0035-4.

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16

Hu, Yi, and Xiaochun Li. "Discrete Fourier restriction associated with Schrödinger equations." Revista Matemática Iberoamericana 30, no. 4 (2014): 1281–300. http://dx.doi.org/10.4171/rmi/815.

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17

Hickman, Jonathan, and Keith M. Rogers. "Improved Fourier restriction estimates in higher dimensions." Cambridge Journal of Mathematics 7, no. 3 (2019): 219–82. http://dx.doi.org/10.4310/cjm.2019.v7.n3.a1.

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18

Hickman, Jonathan, and James Wright. "An abstract $L^2$ Fourier restriction theorem." Mathematical Research Letters 26, no. 1 (2019): 75–100. http://dx.doi.org/10.4310/mrl.2019.v26.n1.a6.

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19

Carro, María, and Salvador Rodríguez. "New results on restriction of Fourier multipliers." Mathematische Zeitschrift 265, no. 2 (April 15, 2009): 417–35. http://dx.doi.org/10.1007/s00209-009-0522-y.

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20

Hong, Guixiang, Xudong Lai, and Liang Wang. "Fourier restriction estimates on quantum Euclidean spaces." Advances in Mathematics 430 (October 2023): 109232. http://dx.doi.org/10.1016/j.aim.2023.109232.

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21

Lakey, Joseph D. "Weighted Restriction for Curves." Canadian Mathematical Bulletin 36, no. 1 (March 1, 1993): 87–95. http://dx.doi.org/10.4153/cmb-1993-013-5.

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AbstractWe prove weighted norm inequalities for the Fourier transform of the formwhere v is a nonnegative weight function on ℝd and ψ: [— 1,1 ] —> ℝd is a nondegenerate curve. Our results generalize unweighted (i.e. v = 1) restriction theorems of M. Christ, and two-dimensional weighted restriction theorems of C. Carton-Lebrun and H. Heinig.

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22

Christ, Francis Michael, and Shuanglin Shao. "Existence of extremals for a Fourier restriction inequality." Analysis & PDE 5, no. 2 (August 27, 2012): 261–312. http://dx.doi.org/10.2140/apde.2012.5.261.

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23

Stovall, Betsy. "Scale-invariant Fourier restriction to a hyperbolic surface." Analysis & PDE 12, no. 5 (January 1, 2019): 1215–24. http://dx.doi.org/10.2140/apde.2019.12.1215.

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24

Hickman, Jonathan. "AN AFFINE FOURIER RESTRICTION THEOREM FOR CONICAL SURFACES." Mathematika 60, no. 2 (December 13, 2013): 374–90. http://dx.doi.org/10.1112/s002557931300020x.

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25

Dendrinos, Spyridon. "Fourier Restriction of H1 Functions on Polynomial Surfaces." Journal of Fourier Analysis and Applications 13, no. 5 (October 2007): 623–41. http://dx.doi.org/10.1007/s00041-006-6055-4.

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26

Mockenhaupt, G. "Salem sets and restriction properties of Fourier transforms." Geometric and Functional Analysis 10, no. 6 (December 2000): 1579–87. http://dx.doi.org/10.1007/pl00001662.

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27

Chen, Xianghong. "A Fourier restriction theorem based on convolution powers." Proceedings of the American Mathematical Society 142, no. 11 (July 21, 2014): 3897–901. http://dx.doi.org/10.1090/s0002-9939-2014-12148-4.

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28

Dendrinos, Spyridon, and James Wright. "Fourier restriction, polynomial curves and a geometric inequality." Comptes Rendus Mathematique 346, no. 1-2 (January 2008): 45–48. http://dx.doi.org/10.1016/j.crma.2007.11.032.

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29

Bloom, Steven, and Gary Sampson. "Weighted spherical restriction theorems for the Fourier transform." Illinois Journal of Mathematics 36, no. 1 (March 1992): 73–101. http://dx.doi.org/10.1215/ijm/1255987608.

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30

Lai, Xudong, and Yong Ding. "A note on the discrete Fourier restriction problem." Proceedings of the American Mathematical Society 146, no. 9 (May 24, 2018): 3839–46. http://dx.doi.org/10.1090/proc/13975.

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31

Foschi, D., and D. Oliveira e Silva. "Some recent progress on sharp Fourier restriction theory." Analysis Mathematica 43, no. 2 (June 2017): 241–65. http://dx.doi.org/10.1007/s10476-017-0306-2.

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32

Chen, Xianghong, and Andreas Seeger. "Convolution Powers of Salem Measures With Applications." Canadian Journal of Mathematics 69, no. 02 (April 2017): 284–320. http://dx.doi.org/10.4153/cjm-2016-019-6.

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AbstractWe study the regularity of convolution powers for measures supported on Salemsets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for α of the form d/n, n = 2, 3, … there exist α-Salem measures for which the L2Fourier restriction theorem holds in the range. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular α-Salem measures, with sharp regularity results forn-fold convolutions for all n ∈ ℕ.

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33

Drury, S. W., and B. P. Marshall. "Fourier restriction theorems for curves with affine and Euclidean arclengths." Mathematical Proceedings of the Cambridge Philosophical Society 97, no. 1 (January 1985): 111–25. http://dx.doi.org/10.1017/s0305004100062654.

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Let M be a smooth manifold in . One may ask whether , the restriction of the Fourier transform of f to M makes sense for every f in . Since, for does not make sense pointwise, it is natural to introduce a measure σ on M and ask for an inequalityfor every f in (say) the Schwartz class. Results of this kind are called restriction theorems. An excellent survey article on this subject is to be found in Tomas[13].

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34

Ramos, Joao. "Low-Dimensional maximal restriction principles for the Fourier transform." Indiana University Mathematics Journal 71, no. 1 (2022): 339–57. http://dx.doi.org/10.1512/iumj.2022.71.8800.

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35

Christ, Michael, and René Quilodrán. "Gaussians rarely extremize adjoint Fourier restriction inequalities for paraboloids." Proceedings of the American Mathematical Society 142, no. 3 (December 23, 2013): 887–96. http://dx.doi.org/10.1090/s0002-9939-2013-11827-7.

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36

Buschenhenke, Stefan, Detlef Müller, and Ana Vargas. "A Fourier restriction theorem for a perturbed hyperbolic paraboloid." Proceedings of the London Mathematical Society 120, no. 1 (August 5, 2019): 124–54. http://dx.doi.org/10.1112/plms.12286.

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37

Spyridon Dendrinos and James Wright. "Fourier restriction to polynomial curves I: a geometric inequality." American Journal of Mathematics 132, no. 4 (2010): 1031–76. http://dx.doi.org/10.1353/ajm.0.0127.

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38

Oberlin, Daniel M. "A uniform estimate for Fourier restriction to simple curves." Proceedings of the American Mathematical Society 137, no. 12 (August 4, 2009): 4227–42. http://dx.doi.org/10.1090/s0002-9939-09-10047-3.

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39

Oberlin, Daniel M. "Fourier restriction for affine arclength measures in the plane." Proceedings of the American Mathematical Society 129, no. 11 (April 2, 2001): 3303–5. http://dx.doi.org/10.1090/s0002-9939-01-06012-9.

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40

Chen, Xianghong, Dashan Fan, and Lifeng Wang. "Restriction of the Fourier Transform to Some Oscillating Curves." Journal of Fourier Analysis and Applications 24, no. 4 (July 5, 2017): 1141–59. http://dx.doi.org/10.1007/s00041-017-9554-6.

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41

Foschi, Damiano. "Global maximizers for the sphere adjoint Fourier restriction inequality." Journal of Functional Analysis 268, no. 3 (February 2015): 690–702. http://dx.doi.org/10.1016/j.jfa.2014.10.015.

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42

Christ, Michael, and Shuanglin Shao. "On the extremizers of an adjoint Fourier restriction inequality." Advances in Mathematics 230, no. 3 (June 2012): 957–77. http://dx.doi.org/10.1016/j.aim.2012.03.020.

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43

De Carli, Laura, Dmitry Gorbachev, and Sergey Tikhonov. "Pitt inequalities and restriction theorems for the Fourier transform." Revista Matemática Iberoamericana 33, no. 3 (2017): 789–808. http://dx.doi.org/10.4171/rmi/955.

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44

Bak, Jong-Guk, and Seheon Ham. "Restriction of the Fourier transform to some complex curves." Journal of Mathematical Analysis and Applications 409, no. 2 (January 2014): 1107–27. http://dx.doi.org/10.1016/j.jmaa.2013.07.073.

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45

Zhang, Yunfeng. "On Fourier restriction type problems on compact Lie groups." Indiana University Mathematics Journal 72, no. 6 (2023): 2631–99. http://dx.doi.org/10.1512/iumj.2023.72.9317.

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46

Drury, S. W., and K. Guo. "Some remarks on the restriction of the Fourier transform to surfaces." Mathematical Proceedings of the Cambridge Philosophical Society 113, no. 1 (January 1993): 153–59. http://dx.doi.org/10.1017/s0305004100075848.

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AbstractFor a class of kernels, we prove the Lp estimate for the exotic Riesz potential, with which a restriction theorem of the Fourier transform to surfaces of half the ambient dimension is proved. A simpler proof of Barcelo's result is given. We also find that it is possible to combine the Hausdorff–Young theorem with the Fefferman–Zygmund method to prove some optimal results on the restriction theorem.

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47

Oberlin, Daniel M. "A Restriction Theorem for a k-Surface in ℝn." Canadian Mathematical Bulletin 48, no. 2 (June 1, 2005): 260–66. http://dx.doi.org/10.4153/cmb-2005-024-9.

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48

Barceló, Bartolomé. "On the restriction of the Fourier transform and Fourier series to circles of lacunary radii." Rendiconti del Circolo Matematico di Palermo 35, no. 3 (September 1986): 330–48. http://dx.doi.org/10.1007/bf02843902.

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49

Rodríguez-López, Salvador. "Restriction results for multilinear multipliers in weighted settings." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 2 (April 2015): 391–409. http://dx.doi.org/10.1017/s0308210513000164.

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We obtain restriction results of de Leeuw’s type for maximal operators defined through multilinear Fourier multipliers of either strong or weak type acting on weighted Lebesgue spaces. We give some applications of our development. In particular, we obtain periodic weighted results for Coifman–Meyer-, Hörmander- and Hörmander–Mihlin-type multilinear multipliers.

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50

Grünrock, Axel, and Sebastian Herr. "The Fourier restriction norm method for the Zakharov-Kuznetsov equation." Discrete & Continuous Dynamical Systems - A 34, no. 5 (2014): 2061–68. http://dx.doi.org/10.3934/dcds.2014.34.2061.

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