Uniform $L^p$ Resolvent Estimates on the Torus

Jonathan Hickman

doi:10.5802/mrr.1

Uniform

L^{p}

Resolvent Estimates on the Torus

Jonathan Hickman ¹

¹ School of Mathematics, The University of Edinburgh, Edinburgh EH9 3JZ, UK

Mathematics Research Reports, Volume 1 (2020), pp. 31-45.

Abstract

A new range of uniform $L^{p}$ resolvent estimates is obtained in the setting of the flat torus, improving previous results of Bourgain, Shao, Sogge and Yao. The arguments rely on the $ℓ^{2}$ -decoupling theorem and multidimensional Weyl sum estimates.

Received: 2019-07-18
Accepted: 2020-02-18
Published online: 2020-06-30

DOI: 10.5802/mrr.1

Classification: 35J05, 35P20, 11P21

Author's affiliations:

Jonathan Hickman ¹

¹ School of Mathematics, The University of Edinburgh, Edinburgh EH9 3JZ, UK

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Jonathan Hickman. Uniform $L^p$ Resolvent Estimates on the Torus. Mathematics Research Reports, Volume 1 (2020), pp. 31-45. doi : 10.5802/mrr.1. https://mrr.centre-mersenne.org/articles/10.5802/mrr.1/

References
Cited by

[BD13] Jean Bourgain; Ciprian Demeter Improved estimates for the discrete Fourier restriction to the higher dimensional sphere, Illinois J. Math., Volume 57 (2013) no. 1, pp. 213-227 http://projecteuclid.org/euclid.ijm/1403534493 | DOI | MR | Zbl

[BD15a] Jean Bourgain; Ciprian Demeter New bounds for the discrete Fourier restriction to the sphere in 4D and 5D, Int. Math. Res. Not. IMRN (2015) no. 11, pp. 3150-3184 | MR | Zbl

[BD15b] Jean Bourgain; Ciprian Demeter The proof of the $l^{2}$ decoupling conjecture, Ann. of Math. (2), Volume 182 (2015) no. 1, pp. 351-389 | DOI | MR | Zbl

[Bou93] J. Bourgain Eigenfunction bounds for the Laplacian on the $n$ -torus, Internat. Math. Res. Notices (1993) no. 3, pp. 61-66 | DOI | MR | Zbl

[Bou97] Jean Bourgain Analysis results and problems related to lattice points on surfaces, Harmonic analysis and nonlinear differential equations (Riverside, CA, 1995) (Contemp. Math.), Volume 208, Amer. Math. Soc., Providence, RI, 1997, pp. 85-109 | DOI | MR | Zbl

[Bou13] J. Bourgain Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, Israel J. Math., Volume 193 (2013) no. 1, pp. 441-458 | DOI | MR | Zbl

[BSSY15] Jean Bourgain; Peng Shao; Christopher D. Sogge; Xiaohua Yao On $L^{p}$ -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys., Volume 333 (2015) no. 3, pp. 1483-1527 | DOI | MR | Zbl

[DSFKS14] David Dos Santos Ferreira; Carlos E. Kenig; Mikko Salo On $L^{p}$ resolvent estimates for Laplace-Beltrami operators on compact manifolds, Forum Math., Volume 26 (2014) no. 3, pp. 815-849 | DOI | MR | Zbl

[Guo12] Jingwei Guo On lattice points in large convex bodies, Acta Arith., Volume 151 (2012) no. 1, pp. 83-108 | DOI | MR | Zbl

[Hla50] Edmund Hlawka Über Integrale auf konvexen Körpern. I, Monatsh. Math., Volume 54 (1950), pp. 1-36 | DOI | MR | Zbl

[KL19] Yehyun Kwon; Sanghyuk. Lee Sharp Resolvent Estimates Outside of the Uniform Boundedness Range, Commun. Math. Phys. (2019) https://link.springer.com/article/10.1007/s00220-019-03536-y | Zbl

[KN91] Ekkehard Krätzel; Werner Georg Nowak Lattice points in large convex bodies, Monatsh. Math., Volume 112 (1991) no. 1, pp. 61-72 | DOI | MR | Zbl

[KN92] Ekkehard Krätzel; Werner Georg Nowak Lattice points in large convex bodies. II, Acta Arith., Volume 62 (1992) no. 3, pp. 285-295 | DOI | MR | Zbl

[KRS87] C. E. Kenig; A. Ruiz; C. D. Sogge Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., Volume 55 (1987) no. 2, pp. 329-347 | DOI | MR | Zbl

[Mül99] Wolfgang Müller Lattice points in large convex bodies, Monatsh. Math., Volume 128 (1999) no. 4, pp. 315-330 | DOI | MR | Zbl

[She01] Zhongwei Shen On absolute continuity of the periodic Schrödinger operators, Internat. Math. Res. Notices (2001) no. 1, pp. 1-31 | DOI | MR | Zbl

[Sog88] Christopher D. Sogge Concerning the $L^{p}$ norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal., Volume 77 (1988) no. 1, pp. 123-138 | DOI | MR | Zbl

[Sog17] Christopher D. Sogge Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, 210, Cambridge University Press, Cambridge, 2017, xiv+334 pages | DOI | MR | Zbl

[Ste93] Elias M. Stein Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993, xiv+695 pages (With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III) | MR | Zbl

[Tao04] Terence Tao Some recent progress on the restriction conjecture, Fourier analysis and convexity (Appl. Numer. Harmon. Anal.), Birkhäuser Boston, Boston, MA, 2004, pp. 217-243 | DOI | MR | Zbl

[TV00] T. Tao; A. Vargas A bilinear approach to cone multipliers. I. Restriction estimates, Geom. Funct. Anal., Volume 10 (2000) no. 1, pp. 185-215 | DOI | MR | Zbl

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