Sharp {$L\sp p$} estimates for some oscillatory integral operators in {$\Bbb R\sp 1$}

Yang, Chan Woo

doi:10.1215/ijm/1258138501

Cited by 11 publications

(14 citation statements)

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“…Inspired by the method used in [18] and [13], we prove our main result by embeding T λ into a family of analytic operators and using complex interpolation. This method requires us to establish the L 2 − L 2 decay estimate as well as H 1 − L 1 boundedness of operators with different amplitude functions.…”

Section: Introductionmentioning

confidence: 93%

“…By using the result in [11], Pan [6] establish the H 1 E − L 1 boundedness for oscillatory singular integral operators, where H 1 E is a modified Hardy space. Later, Yang [18] and Shi [13] developed the method of Pan to get their corresponding H 1 − L 1 and H 1 E − L 1 boundedness results for the oscillatory operators with homogeneous polynomial phase function. In fact, based on these works, the next result can be obtained.…”

Section: H 1 − L 1 Mapping Property Of the Damped Oscillatory Integramentioning

confidence: 99%

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Sharp Lp decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables

Yan

2018

Sci. China Math.

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We obtain the L p decay of oscillatory integral operators T λ with certain homogeneous polynomial phase of degree d in (n + n)-dimensions. In this paper we require that d > 2n. If d/(d − n) < p < d/n, the decay is sharp and the decay rate is related to the Newton distance. In the case of p = d/n or d/(d − n), we also obtain the almost sharp decay, here "almost" means the decay contains a log(λ) term. For otherwise, the L p decay of T λ is also obtained but not sharp. A counterexample also arises in this paper to show that d/(d − n) ≤ p ≤ d/n is not necessary to guarantee the sharp decay.

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Section: Introductionmentioning

confidence: 93%

Section: H 1 − L 1 Mapping Property Of the Damped Oscillatory Integramentioning

confidence: 99%

Sharp Lp decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables

Yan

2018

Sci. China Math.

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“…Moreover, it suffices to show that the above L p estimates are true for extreme points (k, l) ∈ N (S). For further results, one can see Greenleaf-Seeger [10], Yang [35,36] and Shi-Yan [29] for earlier work. For linear and multi-linear estimates, we refer the reader to Carbery-Christ-Wright [1], Carbery-Wright [3], Phong-Stein-Sturm [22], Christ-Li-Tao-Thiele [4] and Gressman-Xiao [12].…”

Section: Introductionmentioning

confidence: 97%

Damping estimates for oscillatory integral operators with real-analytic phases and its applications

Shi

Yan

2019

Forum Mathematicum

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In this paper, we investigate sharp damping estimates for a class of one dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are able to give a new proof of the sharp L p estimates which have been proved by Xiao in Endpoint estimates for one-dimensional oscillatory integral operators, Advances in Mathematics, 316, 255-291 (2017). The damping estimates obtained in this paper are of independent interest.In this section, we shall first give the concept of horizontally (vertically) convex domains which were introduced by Phong-Stein-Sturm [22]; see also for some related remarks. This notion of convexity turns out to be important in this paper. In fact, the operator van der Corput lemma will be established for oscillatory integral operators supported on horizontally and vertically convex domains. Finally, we shall give a variant of Stein-Weiss interpolation with change of measures. Some related results will be also included in this section.

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“…This result was extended to the case of smooth phases by Greenblatt [7]; see Rychkov [19] for a partial result. On the other hand, estimates of T λ on L p were also studied by many authors [9,18,28,29,11,23,21,22,5]. Recently, general sharp L p decay estimates have been proved by Xiao [26].…”

Section: Introductionmentioning

confidence: 98%

Sharp Bounds for Oscillatory Integral Operators with Homogeneous Polynomial Phases

He¹,

Shi²

2019

Can. Math. Bull.

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We obtain sharp L p bounds for oscillatory integral operators with generic homogeneous polynomial phases in several variables. The phases considered in this paper satisfy the rank one condition which is an important notion introduced by Greenleaf, Pramanik and Tang. Under certain additional assumptions, we can establish sharp damping estimates with critical exponents to prove endpoint L p estimates.

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Sharp {$L\sp p$} estimates for some oscillatory integral operators in {$\Bbb R\sp 1$}

Cited by 11 publications

References 0 publications

Sharp Lp decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables

Sharp Lp decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables

Damping estimates for oscillatory integral operators with real-analytic phases and its applications

Sharp Bounds for Oscillatory Integral Operators with Homogeneous Polynomial Phases

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