Sharp estimates for oscillatory integral operators via polynomial partitioning

Guth, Larry; Hickman, Jonathan; Iliopoulou, Marina

doi:10.48550/arxiv.1710.10349

Cited by 13 publications

(33 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof of Proposition 4.7 is a simple exercise and is omitted (see [16] for similar results in the (more complicated) context of oscillatory integral operators).…”

Section: Basic Properties Of the K-broad Normsmentioning

confidence: 99%

“…The next generation of cells is constructed from the varieties which arise from the algebraic case in Theorem 5.1. Fix O j ∈ O j,alg so that there exists a transverse complete intersection Y j of dimension m − 1 and deg (Bj ) (16) where, for convenience, B j := B ∩ N δ Y j . Indeed, provided C tang > 0 is sufficiently large,…”

Section: Finding Polynomial Structurementioning

confidence: 99%

“…For a suitable choice of constant C tang , combining the information in the two previous displays yields (16).…”

Section: Finding Polynomial Structurementioning

confidence: 99%

“…One may refine the above observations to show that α n → Ω −1 as n tends to infinity and so, as 7/4 = 1.75 < Ω −1 , Theorem 1.5 does not constitute an improvement for Conjecture 1.1 in high dimensions. However, by explicitly calculating values of p n , we obtain improvements for Conjecture 1.1 in all dimensions n belonging to the following list: 5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,30,31,32,33,34,35,37,39,40,41,42,44,46,47,48,49,51,53,55,56,58,60,62,65,67,69,72,74,76,81,83, ...…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

New Kakeya estimates using the polynomial Wolff axioms

Hickman,

Rogers

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We obtain new bounds for the Kakeya maximal conjecture in most dimensions n < 100, as well as improved bounds for the Kakeya set conjecture when n = 7 or 9. For this we consider Guth and Zahl's strengthened formulation of the maximal conjecture, concerning families of tubes that satisfy the polynomial Wolff axioms. Our results give improved estimates for this strengthened formulation when n = 5 or n 7.

show abstract

“…The proof of Proposition 4.7 is a simple exercise and is omitted (see [16] for similar results in the (more complicated) context of oscillatory integral operators).…”

Section: Basic Properties Of the K-broad Normsmentioning

confidence: 99%

Section: Finding Polynomial Structurementioning

confidence: 99%

“…For a suitable choice of constant C tang , combining the information in the two previous displays yields (16).…”

Section: Finding Polynomial Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

New Kakeya estimates using the polynomial Wolff axioms

Hickman,

Rogers

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…For d ≥ 3 substantial progresses have been achieved for the last couple of decades but the conjecture still remains open. We refer to [18,7,49,31,8,22] and references therein for details.…”

Section: Introduction and Main Theoremsmentioning

confidence: 99%

Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators

Jeong¹,

Lee²

2019

Preprint

View full text Add to dashboard Cite

We study the maximal estimates for the bilinear spherical average and the bilinear Bochner-Riesz operator. First, we obtain L p × L q → L r estimates for the bilinear spherical maximal function on the optimal range. Thus, we settle the problem which was previously considered by Geba, Greenleaf, Iosevich, Palsson and Sawyer, later Barrionevo, Grafakos, D. He, Honzík and Oliveira, and recently Heo, Hong and Yang. Secondly, we consider L p × L q → L r estimates for the maximal bilinear Bochner-Riesz operators and improve the previous known results. For the purpose we draw a connection between the maximal estimates and the square function estimates for the classical Bochner-Riesz operators.

show abstract

Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

Kwon

Lee

2019

Commun. Math. Phys.

View full text Add to dashboard Cite

In this paper we are concerned with resolvent estimates for the Laplacian ∆ in Euclidean spaces. Uniform resolvent estimates for ∆ were shown by Kenig, Ruiz and Sogge [32] who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp L p -L q bounds depending on z has not been considered in a general framework which admits all possible p, q. In this paper, we present a complete picture of sharp L p -L q resolvent estimates, which may depend on z. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner-Riesz operators of negative index.2010 Mathematics Subject Classification. 35B45, 42B15.

show abstract

Sharp estimates for oscillatory integral operators via polynomial partitioning

Cited by 13 publications

References 0 publications

New Kakeya estimates using the polynomial Wolff axioms

New Kakeya estimates using the polynomial Wolff axioms

Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators

Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

Contact Info

Product

Resources

About