Academic literature on the topic 'Fourier restriction theorems'

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].

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.

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].

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.

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.

Dans cette thèse, on étudie des inégalités de Carleman L^p pour des problèmes elliptiques et leurs applications à la quantification du prolongement unique par rapport aux perturbations du laplacien. On s'intéresse d'abord aux inégalités de Carleman L^p sur une bande de R^d (dgeq 3), notée mathcal{S}:= (0,1) imes R^{d-1}, pour le Laplacien. Grâce à la transformée de Fourier et une factorisation de l'opérateur conjugué, nous réduisons la démonstration de ces inégalités à la construction d'une paramétrice pour le problème du Laplacien avec des conditions au bord.En utilisant cette paramétrice, on redémontre d'abord des inégalités classiques de Carleman L^2 pour le Laplacien. Ensuite, en appliquant des techniques d'analyse harmonique, notamment le théorème de restriction de Fourier pour établir des résultats de continuité de type L^p-L^q , on obtient des estimations L^p - L^q sur cette paramétrice.On applique ensuite ces méthodes au cas qui nous intéresse, à savoir les inégalités de Carleman L^p pour le Laplacien défini sur Omega , un ouvert borné et régulier de R^d (dgeq 3) , avec un second membre f_2 + f_{2 *'} + div F , f_2 in L^2(Omega), , f_{2 *'} in L^{ frac{2d}{d+2}}(Omega), ,F in L^2(Omega; C^{d}), et une condition de Dirichlet g in H^{frac{1}{2}}(partial Omega) . On montre deux estimations de Carleman globales : une sur la norme H^1 de la solution et une sur sa norme L^{frac{2d}{d-2}} , en termes de normes L^2 à poids de f_2 et F , de la norme L^{frac{2d}{d+2}} de f_{2 *'} et de la norme H^{frac{1}{2}} de g . Cela nous permet, par exemple, d'obtenir une quantification du prolongement unique pour les solutions de Delta u = V u + W_1 cdotabla u + div(W_2 u) en fonction des normes de V dans L^{q_0}(Omega) , de W_1 dans L^{q_1}(Omega) et de W_2 dans L^{q_2}(Omega) pour q_0 in (d/2, infty] et q_1 et q_2 satisfaisant soit q_1, , q_2 > (3d-2)/2 et frac{1}{q_1} + frac{1}{q_2}< 4 (1-frac{1}{d})/(3d-2) , soit q_1, , q_2 > 3d/2 .Dans une troisième partie, on étudie une quantification du prolongement unique des solutions de l'équation Delta u = V u + W_1 cdotabla u + div(W_2 u) mais avec des potentiels d'ordre un plus singuliers dans la classe limite d'intégrabilité. En particulier, on considère le cas W_1 in L^{q_1} et W_2 in L^{q_2} , avec q_1>d et q_2 >d . En utilisant le lemme de T. Wolff sur les mesures euclidiennes et une version raffinée des estimations de Carleman, on obtient des résultats de quantification du prolongement unique pour les solutions u de Delta u = V u + W_1 cdotabla u + div (W_2 u) en fonction des normes des potentiels
In this thesis, we study L^p Carleman inequalities for elliptic problems and their applications to the quantification of unique continuation with respect to perturbations of the Laplacian. We first focus on L^p Carleman inequalities on a strip of R^d (dgeq 3) , denoted mathcal{S}:= (0,1) imes R^{d-1} , for the Laplacian. Using the Fourier transform and a factorisation of the conjugate operator, we reduce the proof of these inequalities to the construction of a parametrix for the Laplacian problem with boundary conditions. Utilising this parametrix, we first reprove classical L^2 Carleman inequalities for the Laplacian. Then, applying harmonic analysis techniques, particularly the Fourier restriction theorem to establish L^p-L^q type continuity results, we obtain L^p - L^q estimates for this parametrix.We then apply these methods to the case of interest, namely L^p Carleman inequalities for the Laplacian defined on Omega , a bounded and regular open subset of R^d (d geq 3) , with a right-hand side f_2 + f_{2 *'} + div F , f_2 in L^2(Omega), , f_{2 *'} in L^{ frac{2d}{d+2}}(Omega), ,F in L^2(Omega; C^{d}) , and a Dirichlet condition g in H^{frac{1}{2}}(partial Omega) . We establish two global Carleman estimates: one on the H^1 norm of the solution and another on its L^{frac{2d}{d-2}} norm, in terms of weighted L^2 norms of f_2 and F , the L^{frac{2d}{d+2}} norm of f_{2 *'} , and the H^{frac{1}{2}} norm of g . This allows us, for example, to obtain a quantification of unique continuation for solutions of Delta u = V u + W_1 cdotabla u + div(W_2 u) in terms of the norms of V in L^{q_0}(Omega) , W_1 in L^{q_1}(Omega) , and W_2 in L^{q_2}(Omega) for q_0 in (d/2, infty] and q_1 and q_2 satisfying either q_1, , q_2 > (3d-2)/2 and frac{1}{q_1} + frac{1}{q_2}< 4(1-frac{1}{d})/(3d-2) , or q_1, , q_2 > 3d/2 .In the third part, we study a quantification of unique continuation for solutions of the equation Delta u = V u + W_1 cdotabla u + div(W_2 u) but with first-order potentials that are more singular in the limit integrability class. In particular, we consider the case where W_1 in L^{q_1} and W_2 in L^{q_2} , with q_1 > d and q_2 > d . Using T. Wolff's lemma on Euclidean measures and a refined version of Carleman estimates, we obtain unique continuation quantification results for solutions u of Delta u = V u + W_1 cdotabla u + div(W_2 u) in terms of the norms of the potentials

We study the Fourier restriction phenomenon in settings where there is no underlying proper smooth subvariety. We prove an (Lp, L2) restriction theorem in general locally compact abelian groups and apply it in groups such as (Z/pLZ)n, R and locally compact ultrametric fields K. The problem of existence of Salem sets in a locally compact ultrametric field (K, | · |) is also considered. We prove that for every 0 < α < 1 and ǫ > 0 there exist a set E ⊂ K and a measure μ supported on E such that the Hausdorff dimension of E equals α and |bμ(x)| ≤ C|x|−α 2 +ǫ. We also establish the optimal extension of the Hausdorff-Young inequality in the compact ring of integers R of a locally compact ultrametric field K. We shall prove the following: For every 1 ≤ p ≤ 2 there is a Banach function space Fp(R) with σ-order continuous norm such that (i) Lp(R) ( Fp(R) ( L1(R) for every 1 < p < 2. (ii) The Fourier transform F maps Fp(R) to ℓp′ continuously. (iii) Lp(R) is continuously included in Fp(R) and Fp(R) is continuously included in L1(R). (iv) If Z is a Banach function space with the same properties as Fp(R) above, then Z is continuously included in Fp(R). (v) F1(R) = L1(R) and F2(R) = L2(R).

Abstract There are several possible extensions of this theorem. For example, one may use other than rectangular lattices, which in general are more efficient (see e.g. Peterson and Middleton 1962; Mersereau 1979; Mersereau and Speake 1983; Stark 1979; Higgins 1985; Butzer and Hinsen 1989; Higgins 1996, Chapter 14). However, in any case, bandlimitation is a rather restrictive condition, since a band-limited function is in particular an entire function which cannot be simultaneously duration-limited, in view of the uniqueness theorem for analytic functions.

This chapter mostly considers the domains of type Dsubscript (l), which are in some sense “closest” to the principal root jet, since it turns out that the other domains Dsubscript (l) with l ≥ 2 are easier to handle. In a first step, by means of some lower bounds on the r-height, this chapter establishes favorable restriction estimates in most situations, with the exception of certain cases where m = 2 and B = 3 or B = 4. In some cases the chapter applies interpolation arguments in order to capture the endpoint estimates for p = psubscript c. Sometimes this can be achieved by means of a variant of the Fourier restriction theorem. However, in most of these cases the chapter applies complex interpolation in a similar way as has been done in Chapter 5.

This chapter turns to the proof of a proposition from the previous chapter. Given the operators appearing in that proposition, this chapter establishes the endpoint result thereof by means of Stein's interpolation theorem for analytic families of operators. It constructs analytic families of complex measure μ‎subscript Greek small letter zeta, for ζ‎ in the complex strip Σ‎ given by 0 ≤ Reζ‎ ≤ 1, by introducing complex coefficients in the sums defining the measures ν‎subscript Greek small letter delta,jsuperscript V and ν‎subscript Greek small letter delta,jsuperscript V I, respectively. These coefficients are chosen as exponentials of suitable affine-linear expression in ζ‎ in such a way that, in particular, μ‎subscript Greek small letter theta subscript c = ν‎subscript Greek small letter delta,jsuperscript V I, respectively, μ‎subscript Greek small letter theta subscript c = ν‎subscript Greek small letter delta,jsuperscript V I. As it turns out, the main problem consists in establishing suitable uniform bounds for the measure μ‎subscript Greek small letter zeta when ζ‎ lies on the right boundary line of Σ‎.

Many users of coherent imaging techniques would be delighted with a lens that produces not only a high quality Fourier transform but also a high resolution image of a specified object plane. Unfortunately there is a theorem in geometrical optics that prohibits the design of lens systems that can perform both these tasks flawlessly. To shed some light on the nature of this limitation several proofs of this theorem are presented. One of these proofs, based on a straightforward application of the wave equation without any reference to geometrical optics, shows clearly that this design restriction has a firm foundation in the basic laws of physics.