Two problems associated with convex finite type domains

Iosevich, Alex; Sawyer, Eric T.; Seeger, Andreas

doi:10.5565/publmat_46102_09

Cited by 10 publications

(10 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Collecting the estimates (18), (19) and (20) we have Hence, applying the Hausdorff-Young inequality to (1) with 2 p < 2d/ (d − 1) and 1/p + 1/q = 1, we obtain…”

Section: Let Us Show That the Matrixmentioning

confidence: 99%

Discrepancy for convex bodies with isolated flat points

et al. 2020

View full text Add to dashboard Cite

We consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary. We estimate the L p norm of the discrepancy with respect to the translation variable as the dilation parameter goes to infinity. If there is a single flat point with normal in a rational direction we obtain an asymptotic expansion for this norm. Anomalies may appear when two flat points have opposite normals. When all the flat points have normals in generic irrational directions, we obtain a smaller discrepancy. Our proofs depend on careful estimates for the Fourier transform of the characteristic function of the convex body.(3)1991 Mathematics Subject Classification. 11H06, 42B05.

show abstract

“…Collecting the estimates (18), (19) and (20) we have Hence, applying the Hausdorff-Young inequality to (1) with 2 p < 2d/ (d − 1) and 1/p + 1/q = 1, we obtain…”

Section: Let Us Show That the Matrixmentioning

confidence: 99%

Discrepancy for convex bodies with isolated flat points

et al. 2020

View full text Add to dashboard Cite

show abstract

“…2] and [7, 1.1], respectively). However, the conclusion of Lemma 1.1 which we may draw from Theorem 1.1 in [7] is now much weaker than before: this is so because condition (2.8), which is necessary for every p > 2, is still only sufficient for p ≥ 4, independently of n.…”

mentioning

confidence: 86%

Variations on Bochner–Riesz multipliers in the plane

Debertol

2006

Studia Math.

View full text Add to dashboard Cite

Abstract. We consider the multiplier m µ defined for ξ ∈ R 2 byshow that the optimal range of µ's for which m µ is a Fourier multiplier on L p is the same as for Bochner-Riesz means. The key ingredient is a lemma about some modifications of Bochner-Riesz means inside convex regions with smooth boundary and non-vanishing curvature, providing a more flexible version of a result by Iosevich et al. [Publ. Mat. 46 (2002)]. As an application, we show that the same characterization also holds true for the multiplier p µ (ξ) .2 . Finally, we briefly discuss the n-dimensional analogue of these results.

show abstract

“…Remark 2.4. If A is a fixed matrix, the above result is given directly by [9,Theorem 1.3]. For our need, A is allowed to change.…”

Section: Domain Which Contains the Origin As An Inner Point With Smooth Boundary Of Finite Type If T/amentioning

confidence: 99%

“…Remark 1.2. The key to prove this theorem is an application of a delicate estimate of the Fourier transform of surface carried measure obtained in Iosevich, Sawyer and Seeger [9]. Our main goal was to weaken curvature assumptions in high dimensions, namely to extend the results in [13,8,7] to arbitrary finite type domains.…”

Section: Introductionmentioning

confidence: 99%

Lattice points in stretched model domains of finite type in Rd

Guo

Wang

2018

Journal of Number Theory

View full text Add to dashboard Cite

We study an optimal stretching problem for certain convex domain in R d (d ≥ 3) whose boundary has points of vanishing Gaussian curvature. We prove that the optimal domain which contains the most positive (or least nonnegative) lattice points is asymptotically balanced. This type of problem has its origin in the "eigenvalue optimization among rectangles" problem in spectral geometry. Our proof relies on two-term bounds for lattice counting for general convex domains in R d and an explicit estimate of the Fourier transform of the characteristic function associated with the specific domain under consideration.

show abstract

Two problems associated with convex finite type domains

Cited by 10 publications

References 28 publications

Discrepancy for convex bodies with isolated flat points

Discrepancy for convex bodies with isolated flat points

Variations on Bochner–Riesz multipliers in the plane

Lattice points in stretched model domains of finite type in Rd

Contact Info

Product

Resources

About