Fourier decay of fractal measures on hyperboloids

Barron, Alex; Erdoğan, M. Burak; Harris, Terence L. J.

doi:10.1090/tran/8283

Cited by 6 publications

(13 citation statements)

References 39 publications

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“…, d − 1 (their notation) = n (ours); see also Theorem 2 and Proposition 2.1 of [3]. When D 2 P is positive definite, we can complement Corollary 1.3 with Du and Zhang's deep result in Theorem 2.4 of [18], so that…”

Section: Introductionmentioning

confidence: 88%

“…where we assume that 1 ≤ m ≤ n/2 without loss of generality. These symbols were also extensively studied in [3], where several positive and negative results were established. By (2), we only need to work with α > n/2.…”

Section: (Iii)mentioning

confidence: 99%

“…When P is a non-elliptic quadratic symbol, this result was already proved in Theorem 1, eq. ( 4) of [3]. To compare their results with ours, the reader can use the identity…”

Section: Introductionmentioning

confidence: 97%

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Pointwise convergence over fractals for dispersive equations with homogeneous symbol

Eceizabarrena,

Ponce-Vanegas

2021

Preprint

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We study the problem of pointwise convergence for equations of the type i ∂tu + P (D)u = 0, where the symbol P is real, homogeneous and non-singular. We prove that for initial data f ∈ H s (R n ) with s > (n − α + 1)/2 the solution u converges to f H α -a.e, where H α is the αdimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of the symbol. On the other hand, we prove negative results for a large family of polynomial symbols P . Given α, we exploit a Talbot-like effect to construct regular initial data whose solutions u diverge in sets of Hausdorff dimension α. We also construct data for quadratic symbols like the saddle to show that our positive results are sometimes best possible. To compute the dimension of the sets of divergence we use a Mass Transference Principle from Diophantine approximation theory.

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Section: Introductionmentioning

confidence: 88%

Section: (Iii)mentioning

confidence: 99%

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Pointwise convergence over fractals for dispersive equations with homogeneous symbol

Eceizabarrena,

Ponce-Vanegas

2021

Preprint

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“…I am not aware of such results. However, in addition to the spherical averages (discussed in Section 4), which have been studied for a long time, there are recent estimates for cones and hyperboloids; see [2], [13], and [1].…”

Section: Product Setsmentioning

confidence: 99%

Hausdorff dimension and projections related to intersections

Mattila¹

2022

Publicacions Matemàtiques

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“…where L : R n−1 → R n−1 is an invertible, self-adjoint linear mapping of signature σ . This prototypical case is essentially treated in [2] (see also [11]) but, for completeness, the details are given.…”

Section: Constant Coefficient Decoupling: Quadratic Casementioning

confidence: 99%

Sharp $$L^p$$ estimates for oscillatory integral operators of arbitrary signature

Hickman

Iliopoulou

2022

Math. Z.

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The sharp range of $$L^p$$ L p -estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a general signature assumption on the phase. This simultaneously generalises earlier work of the authors and Guth, which treats the maximal signature case, and also work of Stein and Bourgain–Guth, which treats the minimal signature case.

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Fourier decay of fractal measures on hyperboloids

Cited by 6 publications

References 39 publications

Pointwise convergence over fractals for dispersive equations with homogeneous symbol

Pointwise convergence over fractals for dispersive equations with homogeneous symbol

Hausdorff dimension and projections related to intersections

Sharp $$L^p$$ estimates for oscillatory integral operators of arbitrary signature

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