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Journal articles on the topic 'Fourier restriction theorems'

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Published: 25 January 2025

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1

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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2

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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3

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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4

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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5

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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6

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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7

Ferreyra, Elida, and Marta Urciuolo. "Restriction Theorems for Anisotropically Homogeneous Hypersurfaces of." gmj 15, no. 4 (December 2008): 643–51. http://dx.doi.org/10.1515/gmj.2008.643.

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Abstract For and β 1, . . . , β 𝑛 > 1 let be defined by , let 𝐵 be the open unit ball in and let ∑ = {(𝑥, φ (𝑥)) : 𝑥 ∈ 𝐵}. For let 𝑅𝑓 : ∑ → ℂ be defined by where denotes the usual Fourier transform of 𝑓. Let σ be the Borel measure on ∑ defined by σ (𝐴) = ∫𝐵 χ 𝐴 (𝑥, φ (𝑥)) 𝑑𝑥 and 𝐸 be the type set for the operator 𝑅, i.e. the set of pairs for which there exists 𝑐 > 0 such that for all . In this paper we obtain a polygonal domain contained in 𝐸. We also give necessary conditions for a pair . In some cases this result is sharp up to endpoints.

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8

Ferreyra, E., T. Godoy, and M. Urciuolo. "Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in R3." Studia Mathematica 160, no. 3 (2004): 249–65. http://dx.doi.org/10.4064/sm160-3-4.

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9

Fraser, Robert, and Kyle Hambrook. "Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in the p-adic numbers." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 3 (January 29, 2019): 1265–88. http://dx.doi.org/10.1017/prm.2018.115.

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AbstractWe exhibit the first explicit examples of Salem sets in ℚp of every dimension 0 < α < 1 by showing that certain sets of well-approximable p-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the p-adic setting.

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10

Cluckers, Raf. "Analytic van der Corput Lemma for p-adic and Fq((t)) oscillatory integrals, singular Fourier transforms, and restriction theorems." Expositiones Mathematicae 29, no. 4 (2011): 371–86. http://dx.doi.org/10.1016/j.exmath.2011.06.004.

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11

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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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

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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14

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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15

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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16

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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17

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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18

Gupta, Sanjiv Kumar. "Generalized De Leeuw Theorem." gmj 12, no. 1 (March 2005): 89–96. http://dx.doi.org/10.1515/gmj.2005.89.

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19

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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20

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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21

Oberlin, Daniel M. "A uniform Fourier restriction theorem for surfaces in $\mathbb {R}^{d}$." Proceedings of the American Mathematical Society 140, no. 1 (June 29, 2011): 263–65. http://dx.doi.org/10.1090/s0002-9939-2011-11218-8.

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22

Oberlin, Daniel M. "A uniform Fourier restriction theorem for surfaces in $\mathbb {R}^{3}$." Proceedings of the American Mathematical Society 132, no. 4 (October 15, 2003): 1195–99. http://dx.doi.org/10.1090/s0002-9939-03-07289-7.

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23

Vitturi, Marco. "A note on maximal Fourier restriction for spheres in all dimensions." Glasnik Matematicki 57, no. 2 (December 30, 2022): 313–19. http://dx.doi.org/10.3336/gm.57.2.10.

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We prove a maximal Fourier restriction theorem for hypersurfaces in \(\mathbb{R}^{d}\) for any dimension \(d\geq 3\) in a restricted range of exponents given by the Tomas-Stein theorem (spheres being the most canonical example). The proof consists of a simple observation. When \(d=3\) the range corresponds exactly to the full Tomas-Stein one, but is otherwise a proper subset when \(d>3\). We also present an application regarding the Lebesgue points of functions in \(\mathcal{F}(L^p)\) when \(p\) is sufficiently close to 1.

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24

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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25

Buschenhenke, Stefan, Detlef Müller, and Ana Vargas. "A Fourier restriction theorem for a two-dimensional surface of finite type." Analysis & PDE 10, no. 4 (May 9, 2017): 817–91. http://dx.doi.org/10.2140/apde.2017.10.817.

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26

Kumar, Pratyoosh. "Fourier restriction theorem and characterization of weakL2eigenfunctions of the Laplace–Beltrami operator." Journal of Functional Analysis 266, no. 9 (May 2014): 5584–97. http://dx.doi.org/10.1016/j.jfa.2013.10.009.

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27

Chen, Xianghong. "Sets of Salem type and sharpness of the $L^2$-Fourier restriction theorem." Transactions of the American Mathematical Society 368, no. 3 (June 17, 2015): 1959–77. http://dx.doi.org/10.1090/tran/6396.

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28

Arendt, Wolfgang, and Shangquan Bu. "OPERATOR-VALUED FOURIER MULTIPLIERS ON PERIODIC BESOV SPACES AND APPLICATIONS." Proceedings of the Edinburgh Mathematical Society 47, no. 1 (February 2004): 15–33. http://dx.doi.org/10.1017/s0013091502000378.

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AbstractLet $1\leq p,q\leq\infty$, $s\in\mathbb{R}$ and let $X$ be a Banach space. We show that the analogue of Marcinkiewicz’s Fourier multiplier theorem on $L^p(\mathbb{T})$ holds for the Besov space $B_{p,q}^s(\mathbb{T};X)$ if and only if $1\ltp\lt\infty$ and $X$ is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann’s result (Math. Nachr.186 (1997), 5–56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.AMS 2000 Mathematics subject classification: Primary 47D06; 42A45

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29

Oberlin, Daniel, and Richard Oberlin. "Application of a Fourier Restriction Theorem to Certain Families of Projections in $${\mathbb {R}}^3$$ R 3." Journal of Geometric Analysis 25, no. 3 (March 22, 2014): 1476–91. http://dx.doi.org/10.1007/s12220-014-9480-7.

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30

Hamid, Ashwaq Q., and Burak Abedulhadi. "DESIGN AND EVALUATION OF A WEB BASED VIRTUAL DSP LABORATORY USING GUI AND HTML." Journal of Engineering 17, no. 02 (March 1, 2011): 279–306. http://dx.doi.org/10.31026/j.eng.2011.02.07.

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ABSTRACT:Engineering education should involve practical laboratory to support theoretical foundation and developstudent skills. These hands on laboratories have some disadvantages such as expensive, supervision required, time and place restrictions. This paper presents design and evaluation of a web based virtual laboratory for teaching Digital Signal Processing (DSP) to undergraduate students in Electromechanical Engineering Department at the University of Technology. The laboratory experiments includes classification of signals, sampling theorem, Fourier series, complex Fourier series, Fourier transform, inverse Fourier transform, discrete Fourier transform, Fast Fourier Transform (FFT),convolution , Z-transform, and digital filters. Graphical User Interface (GUI) feature of MATLAB have been used to provide students with a friendly and visual approach in specifying input parameters while Hyper Text Markup Language (HTML) was used to illustrate theoretical foundations. The questionnaire survey and five point Likert scale are utilized in performing evaluation. Results of this evaluation showed that the proposed virtual DSP laboratory was helped students in DSP concepts, made positive effects on students’ achievements and attitudes when compared to traditional teaching methods.

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31

Buschenhenke, Stefan, Detlef Müller, and Ana Vargas. "Partitions of Flat One-Variate Functions and a Fourier Restriction Theorem for Related Perturbations of the Hyperbolic Paraboloid." Journal of Geometric Analysis 31, no. 7 (February 18, 2021): 6941–86. http://dx.doi.org/10.1007/s12220-020-00587-9.

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32

Buschenhenke, Stefan. "A sharp $$L^p-L^q$$ L p - L q -Fourier restriction theorem for a conical surface of finite type." Mathematische Zeitschrift 280, no. 1-2 (March 25, 2015): 367–99. http://dx.doi.org/10.1007/s00209-015-1429-4.

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33

Saﬁna, R. M. "Keldysh problem for Pulkin’s equation in a rectangular domain." Vestnik of Samara University. Natural Science Series 21, no. 3 (May 19, 2017): 53–63. http://dx.doi.org/10.18287/2541-7525-2015-21-3-53-63.

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In this article for the mixed type equation with a singular coeﬃcient Keldysh problem of incomplete boundary conditions is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral prob- lem the criterion of uniqueness is established. The solution is constructed as the summary of Fourier-Bessel row. At the foundation of the uniform convergence of a row there is a problem of small denominators.Under some restrictions on these tasks evaluation of separation from zero of a small denominator with the corresponding asymptotics was found, which helped to prove the uniform con- vergence and its derivatives up to the second order inclusive, and the existence theorem in the class of regular solutions.

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34

Antoniou, I., and S. A. Shkarin. "Decay measures on locally compact abelian topological groups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 6 (December 2001): 1257–73. http://dx.doi.org/10.1017/s0308210500001384.

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We show that the Banach space M of regular σ-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces MD and MND, where MD is the set of measures μ ∈ M whose Fourier transform vanishes at infinity and MND is the set of measures μ ∈ M such that ν ∉ MD for any ν ∈ M {0} absolutely continuous with respect to the variation |μ|. For any corresponding decomposition μ = μD + μND (μD ∈ MD and μND ∈ MND) there exist a Borel set A = A(μ) such that μD is the restriction of μ to A, therefore the measures μD and μND are singular with respect to each other. The measures μD and μND are real if μ is real and positive if μ is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from MD and MND.

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35

Frønsdal, Christian. "Relativistic thermodynamics, a Lagrangian field theory for general flows including rotation." International Journal of Geometric Methods in Modern Physics 14, no. 02 (January 18, 2017): 1750017. http://dx.doi.org/10.1142/s0219887817500177.

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Any theory that is based on an action principle has a much greater predictive power than one that does not have such a formulation. The formulation of a dynamical theory of General Relativity, including matter, is here viewed as a problem of coupling Einstein’s theory of pure gravity to an independently chosen and well-defined field theory of matter. It is well known that this is accomplished in a most natural way when both theories are formulated as relativistic, Lagrangian field theories, as is the case with Einstein–Maxwell theory. Special matter models of this type have been available; here a more general thermodynamical model that allows for vortex flows is presented. In a wider context, the problem of subjecting hydrodynamics and thermodynamics to an action principle is one that has been pursued for at least 150 years. A solution to this problem has been known for some time, but only under the strong restriction to potential flows. A variational principle for general flows has become available. It represents a development of the Navier–Stokes–Fourier approach to fluid dynamics. The principal innovation is the recognition that two kinds of flow velocity fields are needed, one the gradient of a scalar field and the other the time derivative of a vector field, the latter closely associated with vorticity. In the relativistic theory that is presented here, the latter is the Hodge dual of an exact 3-form, well known as the notoph field of Ogievetskij and Palubarinov, the [Formula: see text]-field of Kalb and Ramond and the vorticity field of Lund and Regge. The total number of degrees of freedom of a unary system, including the density and the two velocity fields is 4, as expected — as in classical hydrodynamics. In this paper, we do not reduce Einstein’s dynamical equation for the metric to phenomenology, which would have denied the relevance of any intrinsic dynamics for the matter sector, nor do we abandon the equation of continuity - the very soul of hydrodynamics.

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36

Nikolaev, Oleksii, and Mariia Skitska. "The method of determining optimal control of the thermoelastic state of piece-homogeneous body using a stationary temperature field." Radioelectronic and Computer Systems 2024, no. 2 (April 23, 2024): 98–119. http://dx.doi.org/10.32620/reks.2024.2.09.

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This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequences. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demonstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries.

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Bahouri, Hajer, Davide Barilari, and Isabelle Gallagher. "Strichartz Estimates and Fourier Restriction Theorems on the Heisenberg Group." Journal of Fourier Analysis and Applications 27, no. 2 (March 11, 2021). http://dx.doi.org/10.1007/s00041-021-09822-5.

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38

Dabra, Arvish, and N. Shravan Kumar. "Restriction Theorems for the p-Analog of the Fourier–Stieltjes Algebra." Results in Mathematics 79, no. 6 (August 26, 2024). http://dx.doi.org/10.1007/s00025-024-02263-8.

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39

CASPERS, MARTIJN, JAVIER PARCET, MATHILDE PERRIN, and ÉRIC RICARD. "NONCOMMUTATIVE DE LEEUW THEOREMS." Forum of Mathematics, Sigma 3 (October 1, 2015). http://dx.doi.org/10.1017/fms.2015.23.

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Let $\text{H}$ be a subgroup of some locally compact group $\text{G}$. Assume that $\text{H}$ is approximable by discrete subgroups and that $\text{G}$ admits neighborhood bases which are almost invariant under conjugation by finite subsets of $\text{H}$. Let $m:\text{G}\rightarrow \mathbb{C}$ be a bounded continuous symbol giving rise to an $L_{p}$-bounded Fourier multiplier (not necessarily completely bounded) on the group von Neumann algebra of $\text{G}$ for some $1\leqslant p\leqslant \infty$. Then, $m_{\mid _{\text{H}}}$ yields an $L_{p}$-bounded Fourier multiplier on the group von Neumann algebra of $\text{H}$ provided that the modular function ${\rm\Delta}_{\text{G}}$ is equal to 1 over $\text{H}$. This is a noncommutative form of de Leeuw’s restriction theorem for a large class of pairs $(\text{G},\text{H})$. Our assumptions on $\text{H}$ are quite natural, and they recover the classical result. The main difference with de Leeuw’s original proof is that we replace dilations of Gaussians by other approximations of the identity for which certain new estimates on almost-multiplicative maps are crucial. Compactification via lattice approximation and periodization theorems are also investigated.

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Caspers, Martijn, Bas Janssens, Amudhan Krishnaswamy-Usha, and Lukas Miaskiwskyi. "Local and multilinear noncommutative de Leeuw theorems." Mathematische Annalen, May 3, 2023. http://dx.doi.org/10.1007/s00208-023-02611-z.

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AbstractLet $$\Gamma < G$$ Γ < G be a discrete subgroup of a locally compact unimodular group G. Let $$m\in C_b(G)$$ m ∈ C b ( G ) be a p-multiplier on G with $$1 \le p < \infty $$ 1 ≤ p < ∞ and let $$T_{m}: L_p({\widehat{G}}) \rightarrow L_p({\widehat{G}})$$ T m : L p ( G ^ ) → L p ( G ^ ) be the corresponding Fourier multiplier. Similarly, let $$T_{m \vert _\Gamma }: L_p({\widehat{\Gamma }}) \rightarrow L_p({\widehat{\Gamma }})$$ T m | Γ : L p ( Γ ^ ) → L p ( Γ ^ ) be the Fourier multiplier associated to the restriction $$m|_{\Gamma }$$ m | Γ of m to $$\Gamma $$ Γ . We show that $$\begin{aligned} c( {{\,\textrm{supp}\,}}( m|_{\Gamma } ) ) \Vert T_{m \vert _\Gamma }: L_p({\widehat{\Gamma }}) \rightarrow L_p({\widehat{\Gamma }}) \Vert \le \Vert T_{m }: L_p({\widehat{G}}) \rightarrow L_p({\widehat{G}}) \Vert , \end{aligned}$$ c ( supp ( m | Γ ) ) ‖ T m | Γ : L p ( Γ ^ ) → L p ( Γ ^ ) ‖ ≤ ‖ T m : L p ( G ^ ) → L p ( G ^ ) ‖ , for a specific constant $$0 \le c(U) \le 1$$ 0 ≤ c ( U ) ≤ 1 that is defined for every $$U \subseteq \Gamma $$ U ⊆ Γ . The function c quantifies the failure of G to admit small almost $$\Gamma $$ Γ -invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, $$c(\Gamma ) =1$$ c ( Γ ) = 1 when G has small almost $$\Gamma $$ Γ -invariant neighbourhoods. Our result thus extends the de Leeuw restriction theorem from Caspers et al. (Forum Math Sigma 3(e21):59, 2015) as well as de Leeuw’s classical theorem (Ann Math 81(2):364–379, 1965). For real reductive Lie groups G we provide an explicit lower bound for c in terms of the maximal dimension d of a nilpotent orbit in the adjoint representation. We show that $$c(B_\rho ^G) \ge \rho ^{-d/4}$$ c ( B ρ G ) ≥ ρ - d / 4 where $$B_\rho ^G$$ B ρ G is the ball of $$g\in G$$ g ∈ G with $$\Vert {{\,\textrm{Ad}\,}}_g \Vert < \rho $$ ‖ Ad g ‖ < ρ . We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear de Leeuw restriction theorem for pairs $$\Gamma <G$$ Γ < G with $$c(\Gamma ) = 1$$ c ( Γ ) = 1 . We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.

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41

Dabra, Arvish, and N. Shravan Kumar. "Correction To: Restriction Theorems for the p-Analog of the Fourier–Stieltjes Algebra." Results in Mathematics 79, no. 8 (November 16, 2024). http://dx.doi.org/10.1007/s00025-024-02304-2.

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42

Fraser, Robert, Kyle Hambrook, and Donggeun Ryou. "Fourier restriction and well-approximable numbers." Mathematische Annalen, November 1, 2024. http://dx.doi.org/10.1007/s00208-024-03000-w.

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AbstractWe use a deterministic construction to prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for dimension $$d=1$$ d = 1 and parameter range $$0 < a,b \le d$$ 0 < a , b ≤ d and $$b\le 2a$$ b ≤ 2 a . Previous constructions by Hambrook and Łaba [15] and Chen [5] required randomness and only covered the range $$0 < b \le a \le d=1$$ 0 < b ≤ a ≤ d = 1 . We also resolve a question of Seeger [29] about the Fourier restriction inequality on the sets of well-approximable numbers.

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43

Senapati, P. Jitendra Kumar, Pradeep Boggarapu, Shyam Swarup Mondal, and Hatem Mejjaoli. "Restriction theorem for the Fourier–Dunkl transform I: cone surface." Journal of Pseudo-Differential Operators and Applications 14, no. 1 (December 11, 2022). http://dx.doi.org/10.1007/s11868-022-00499-y.

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44

Buschenhenke, Stefan, Detlef Müller, and Ana Vargas. "A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning." Mathematische Zeitschrift, February 7, 2022. http://dx.doi.org/10.1007/s00209-021-02948-8.

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AbstractWe consider a surface with negative curvature in $${{\mathbb {R}}}^3,$$ R 3 , which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of one-variate perturbations, and we hope that the present paper will further help to pave the way for the study of general perturbations of the saddle by means of the polynomial partitioning method.

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45

Senapati, P. Jitendra Kumar, Pradeep Boggarapu, Shyam Swarup Mondal, and Hatem Mejjaoli. "Restriction Theorem for the Fourier–Dunkl Transform and Its Applications to Strichartz Inequalities." Journal of Geometric Analysis 34, no. 3 (January 13, 2024). http://dx.doi.org/10.1007/s12220-023-01530-4.

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46

Garofalo, Nicola. "Some Inequalities for the Fourier Transform and Their Limiting Behaviour." Journal of Geometric Analysis 34, no. 2 (December 29, 2023). http://dx.doi.org/10.1007/s12220-023-01477-6.

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AbstractWe identify a one-parameter family of inequalities for the Fourier transform whose limiting case is the restriction conjecture for the sphere. Using Stein’s method of complex interpolation, we prove the conjectured inequalities when the target space is $$L^2$$ L 2 , and show that this recovers in the limit the celebrated Tomas-Stein theorem.

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47

Mondal, Shyam Swarup, and Jitendriya Swain. "Restriction theorem for the Fourier–Hermite transform and solution of the Hermite–Schrödinger equation." Advances in Operator Theory 7, no. 4 (July 19, 2022). http://dx.doi.org/10.1007/s43036-022-00208-y.

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48

Thangavelu, Sundaram. "Spherical Means on the Heisenberg Group and a Restriction Theorem for the Symplectic Fourier Transform." Revista Matemática Iberoamericana, 1991, 135–55. http://dx.doi.org/10.4171/rmi/108.

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49

Mondal, Shyam Swarup, and Jitendriya Swain. "Correction: Restriction theorem for the Fourier–Hermite transform and solution of the Hermite–Schrödinger equation." Advances in Operator Theory 8, no. 3 (June 14, 2023). http://dx.doi.org/10.1007/s43036-023-00276-8.

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50

Oliveira e Silva, Diogo. "The endpoint Stein–Tomas inequality: old and new." São Paulo Journal of Mathematical Sciences, April 22, 2024. http://dx.doi.org/10.1007/s40863-024-00422-x.

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AbstractThe Stein–Tomas inequality from 1975 is a cornerstone of Fourier restriction theory. Despite its respectable age, it is a fertile ground for current research. This note is centered around three classical applications – to Strichartz inequalities, Salem sets and Roth’s theorem in the primes – and three recent improvements: the sharp endpoint Stein–Tomas inequality in three space dimensions, maximal and variational refinements, and the symmetric Stein–Tomas inequality with applications.

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