doi: 10.3934/cpaa.2022079
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Fourier transform of surface–carried measures of two-dimensional generic surfaces and applications

1. 

Loughborough University, Department of Mathematical Sciences, Loughborough, Leicestershire, LE11 3TU United Kingdom

2. 

Karlsruher Institut für Technologie, Fakultät für Mathematik, Institut für Analysis, Englerstrasse 2, 76131 Karlsruhe, Germany

* Corresponding author

Received  January 2022 Revised  March 2022 Early access April 2022

Fund Project: The second author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Project-ID 258734477–SFB 1173

We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. We also obtain new results on the spectral and scattering theory of discrete Schrödinger operators on the cubic lattice.

Citation: Jean-Claude Cuenin, Robert Schippa. Fourier transform of surface–carried measures of two-dimensional generic surfaces and applications. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022079
References:
[1]

T. AlazardN. Burq and C. Zuily, A stationary phase type estimate, Proc. Amer. Math. Soc., 145 (2017), 2871-2880.  doi: 10.1090/proc/13199.

[2]

V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012.

[3]

M. Ben-ArtziH. Koch and J.-C. Saut, Dispersion estimates for third order equations in two dimensions, Commun. Partial Differ. Equ., 28 (2003), 1943-1974.  doi: 10.1081/PDE-120025491.

[4]

J. Bourgain, Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena, in Geometric aspects of functional analysis, pages 70–98. Springer, Berlin, 2003. doi: 10.1007/978-3-540-36428-3_7.

[5]

J. Bourgain, On random Schrödinger operators on $\Bbb Z^2$, Discrete Contin. Dyn. Syst., 8 (2002), 1-15.  doi: 10.3934/dcds.2002.8.1.

[6] Th. Bröcker, Differentiable Germs and Catastrophes, Cambridge University Press, Cambridge-New York-Melbourne, 1975. 
[7]

J.-C. Cuenin, From spectral cluster to uniform resolvent estimates on compact manifolds, arXiv: 2011.07254.

[8]

J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Commun. Pure Appl. Math., 27 (1974), 207-281.  doi: 10.1002/cpa.3160270205.

[9]

L. ErdősM. Salmhofer and H. T. Yau, Quantum diffusion for the Anderson model in the scaling limit, Ann. Henri Poincaré, 8 (2007), 621-685.  doi: 10.1007/s00023-006-0318-0.

[10]

L. Erdos and M. Salmhofer, Decay of the Fourier transform of surfaces with vanishing curvature, Math. Z., 257 (2007), 261-294.  doi: 10.1007/s00209-007-0125-4.

[11]

M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973.

[12]

A. Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J., 30 (1981), 519-537.  doi: 10.1512/iumj.1981.30.30043.

[13]

I. A. IkromovM. Kempe and D. Müller, Estimates for maximal functions associated with hypersurfaces in $\Bbb R^3$ and related problems of harmonic analysis, Acta Math., 204 (2010), 151-271.  doi: 10.1007/s11511-010-0047-6.

[14]

I. A. Ikromov and D. Müller, On adapted coordinate systems, Trans. Amer. Math. Soc., 363 (2011), 2821-2848.  doi: 10.1090/S0002-9947-2011-04951-2.

[15]

I A. Ikromov and D. Müller, Uniform estimates for the Fourier transform of surface carried measures in $\Bbb R^3$ and an application to Fourier restriction, J. Fourier Anal. Appl., 17 (2011), 1292-1332.  doi: 10.1007/s00041-011-9191-4.

[16]

I A. Ikromov and D. Müller, Fourier restriction for hypersurfaces in three dimensions and Newton polyhedra, in Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2016. doi: 10.1515/9781400881246.

[17]

V. N. Karpushkin, A theorem on uniform estimates for oscillatory integrals with a phase depending on two variables, Trudy Sem. Petrovsk., 238 (1984), 150-169. 

[18]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. 

[19]

Evgeny L. Korotyaev and Jacob Schach Møller, Weighted estimates for the Laplacian on the cubic lattice, Ark. Mat., 57 (2019), 397-428.  doi: 10.4310/ARKIV.2019.v57.n2.a8.

[20]

Y. Kwon and S. Lee, Sharp resolvent estimates outside of the uniform boundedness range, Commun. Math. Phys., 374 (2020), 1417-1467.  doi: 10.1007/s00220-019-03536-y.

[21]

W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc., 69 (1963), 766-770.  doi: 10.1090/S0002-9904-1963-11025-3.

[22]

R. Mandel and R. Schippa, Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces, Ann. Henri Poincaré, 415-445, 2021.

[23]

S. Oh and S. Lee, Uniform stationary phase estimate with limited smoothness, arXiv: 2012.12572.

[24]

L. Palle, Mixed norm Strichartz-type estimates for hypersurfaces in three dimensions, Math. Z., 297 (2021), 1529-1599.  doi: 10.1007/s00209-020-02568-8.

[25]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, Harcourt Brace Jovanovich Publishers, New York, London, 1978.

[26]

W. SchlagC. Shubin and T. Wolff, Frequency concentration and location lengths for the Anderson model at small disorders, J. Anal. Math., 88 (2002), 173-220.  doi: 10.1007/BF02786577.

[27]

C. D. Sogge, Fourier integrals in classical analysis, in Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2017. doi: 10.1017/9781316341186.

[28]

E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, in Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993.

[29]

K. Taira, Limiting absorption principle on $L^p$-spaces and scattering theory, J. Math. Phys., 61 (2020), 092106, 28 pp. doi: 10.1063/5.0011805.

[30]

K. Taira, Uniform resolvent estimates for the discrete Schrödinger operator in dimension three, J. Spectr. Theory, 11 (2021), 1831-1855.  doi: 10.4171/jst/387.

[31]

A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals, Funct. Anal. Appl., 10 (1976), 175-196. 

[32]

R. Vershynin, High-dimensional probability, in Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2018. doi: 10.1017/9781108231596.

[33]

H. Whitney, On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane, Ann. Math., 62 (1955), 374-410.  doi: 10.2307/1970070.

[34]

D. R. Yafaev, Mathematical scattering theory, in Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992. doi: 10.1090/mmono/105.

show all references

References:
[1]

T. AlazardN. Burq and C. Zuily, A stationary phase type estimate, Proc. Amer. Math. Soc., 145 (2017), 2871-2880.  doi: 10.1090/proc/13199.

[2]

V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012.

[3]

M. Ben-ArtziH. Koch and J.-C. Saut, Dispersion estimates for third order equations in two dimensions, Commun. Partial Differ. Equ., 28 (2003), 1943-1974.  doi: 10.1081/PDE-120025491.

[4]

J. Bourgain, Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena, in Geometric aspects of functional analysis, pages 70–98. Springer, Berlin, 2003. doi: 10.1007/978-3-540-36428-3_7.

[5]

J. Bourgain, On random Schrödinger operators on $\Bbb Z^2$, Discrete Contin. Dyn. Syst., 8 (2002), 1-15.  doi: 10.3934/dcds.2002.8.1.

[6] Th. Bröcker, Differentiable Germs and Catastrophes, Cambridge University Press, Cambridge-New York-Melbourne, 1975. 
[7]

J.-C. Cuenin, From spectral cluster to uniform resolvent estimates on compact manifolds, arXiv: 2011.07254.

[8]

J. J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Commun. Pure Appl. Math., 27 (1974), 207-281.  doi: 10.1002/cpa.3160270205.

[9]

L. ErdősM. Salmhofer and H. T. Yau, Quantum diffusion for the Anderson model in the scaling limit, Ann. Henri Poincaré, 8 (2007), 621-685.  doi: 10.1007/s00023-006-0318-0.

[10]

L. Erdos and M. Salmhofer, Decay of the Fourier transform of surfaces with vanishing curvature, Math. Z., 257 (2007), 261-294.  doi: 10.1007/s00209-007-0125-4.

[11]

M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973.

[12]

A. Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J., 30 (1981), 519-537.  doi: 10.1512/iumj.1981.30.30043.

[13]

I. A. IkromovM. Kempe and D. Müller, Estimates for maximal functions associated with hypersurfaces in $\Bbb R^3$ and related problems of harmonic analysis, Acta Math., 204 (2010), 151-271.  doi: 10.1007/s11511-010-0047-6.

[14]

I. A. Ikromov and D. Müller, On adapted coordinate systems, Trans. Amer. Math. Soc., 363 (2011), 2821-2848.  doi: 10.1090/S0002-9947-2011-04951-2.

[15]

I A. Ikromov and D. Müller, Uniform estimates for the Fourier transform of surface carried measures in $\Bbb R^3$ and an application to Fourier restriction, J. Fourier Anal. Appl., 17 (2011), 1292-1332.  doi: 10.1007/s00041-011-9191-4.

[16]

I A. Ikromov and D. Müller, Fourier restriction for hypersurfaces in three dimensions and Newton polyhedra, in Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2016. doi: 10.1515/9781400881246.

[17]

V. N. Karpushkin, A theorem on uniform estimates for oscillatory integrals with a phase depending on two variables, Trudy Sem. Petrovsk., 238 (1984), 150-169. 

[18]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. 

[19]

Evgeny L. Korotyaev and Jacob Schach Møller, Weighted estimates for the Laplacian on the cubic lattice, Ark. Mat., 57 (2019), 397-428.  doi: 10.4310/ARKIV.2019.v57.n2.a8.

[20]

Y. Kwon and S. Lee, Sharp resolvent estimates outside of the uniform boundedness range, Commun. Math. Phys., 374 (2020), 1417-1467.  doi: 10.1007/s00220-019-03536-y.

[21]

W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc., 69 (1963), 766-770.  doi: 10.1090/S0002-9904-1963-11025-3.

[22]

R. Mandel and R. Schippa, Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces, Ann. Henri Poincaré, 415-445, 2021.

[23]

S. Oh and S. Lee, Uniform stationary phase estimate with limited smoothness, arXiv: 2012.12572.

[24]

L. Palle, Mixed norm Strichartz-type estimates for hypersurfaces in three dimensions, Math. Z., 297 (2021), 1529-1599.  doi: 10.1007/s00209-020-02568-8.

[25]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, Harcourt Brace Jovanovich Publishers, New York, London, 1978.

[26]

W. SchlagC. Shubin and T. Wolff, Frequency concentration and location lengths for the Anderson model at small disorders, J. Anal. Math., 88 (2002), 173-220.  doi: 10.1007/BF02786577.

[27]

C. D. Sogge, Fourier integrals in classical analysis, in Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2017. doi: 10.1017/9781316341186.

[28]

E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, in Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993.

[29]

K. Taira, Limiting absorption principle on $L^p$-spaces and scattering theory, J. Math. Phys., 61 (2020), 092106, 28 pp. doi: 10.1063/5.0011805.

[30]

K. Taira, Uniform resolvent estimates for the discrete Schrödinger operator in dimension three, J. Spectr. Theory, 11 (2021), 1831-1855.  doi: 10.4171/jst/387.

[31]

A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals, Funct. Anal. Appl., 10 (1976), 175-196. 

[32]

R. Vershynin, High-dimensional probability, in Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2018. doi: 10.1017/9781108231596.

[33]

H. Whitney, On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane, Ann. Math., 62 (1955), 374-410.  doi: 10.2307/1970070.

[34]

D. R. Yafaev, Mathematical scattering theory, in Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992. doi: 10.1090/mmono/105.

Figure 1.  Pentagonal region, within which strong $L^p$-$L^q$-Fourier restriction extension estimates hold
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