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On Fourier integral operators with Hölder-continuous phase. (English) Zbl 1403.35336

Summary: We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a Hölder-type singularity at the origin. We prove boundedness in \(L^1\) with a precise loss of decay depending on the Hölder exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling-Helson theorem for changes of variables with a Hölder singularity at the origin. The continuity in \(L^2\) is studied as well by providing sufficient conditions and relevant counterexamples. The proofs rely on techniques from time-frequency analysis.

MSC:

35S30 Fourier integral operators applied to PDEs
47G10 Integral operators
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[1] Alonso, R. J.; Lods, B., Free cooling and high-energy tails of granular gases with variable restitution coefficient, SIAM J. Math. Anal., 42, 6, 2499-2538, (2010) · Zbl 1419.76684
[2] R. J. Alonso and B. Lods, Sobolev smoothing of the Boltzmann operator for visco-elastic granular gases, Personal communication (2017).
[3] Bényi, A.; Gröchenig, K.; Okoudjou, K. A.; Rogers, L. G., Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246, 2, 366-384, (2007) · Zbl 1120.42010
[4] Beurling, A.; Helson, H., Fourier-Stieltjes transforms with bounded powers, Math. Scand., 1, 120-126, (1953) · Zbl 0050.33004
[5] Bouchut, F.; Desvillettes, L., A proof of the smoothing properties of the positive part of boltzmann’s kernel, Rev. Mat. Iberoamericana, 14, 1, 47-61, (1998) · Zbl 0912.45014
[6] Brilliantov, N. V.; Pöschel, T., Kinetic Theory of Granular Gases, (2004), Oxford University Press, Oxford · Zbl 1155.76386
[7] Cordero, E.; Nicola, F., Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation, J. Funct. Anal., 254, 2, 506-534, (2008) · Zbl 1136.22006
[8] Cordero, E.; Nicola, F., Some new Strichartz estimates for the Schrödinger equation, J. Differential Equations, 245, 7, 1945-1974, (2008) · Zbl 1154.35081
[9] Cordero, E.; Nicola, F.; Rodino, L., On the global boundedness of Fourier integral operators, Ann. Global Anal. Geom., 38, 4, 373-398, (2010) · Zbl 1200.35347
[10] Cordero, E.; Nicola, F.; Rodino, L., Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potentials, Rev. Math. Phys., 27, 1, 33, (2015) · Zbl 1314.35119
[11] Coriasco, S.; Ruzhansky, M., Global \(L^p\) continuity of Fourier integral operators, Trans. Amer. Math. Soc., 366, 5, 2575-2596, (2014) · Zbl 1301.35231
[12] Feichtinger, H. G., Functional Analysis and Approximation, 60, Banach spaces of distributions of wiener’s type and interpolation, 153-165, (1981), Birkhäuser, Basel
[13] Feichtinger, H. G., On a new Segal algebra, Monatsh. Math., 92, 4, 269-289, (1981) · Zbl 0461.43003
[14] Feichtinger, H. G., Functions, Series, Operators, Vol. \(\text{I, II}\), 35, Banach convolution algebras of Wiener type, 509-524, (1983), North-Holland, Amsterdam
[15] Feichtinger, H. G., Atomic characterizations of modulation spaces through Gabor-type representations, Rocky Mountain J. Math., 19, 1, 113-125, (1989) · Zbl 0780.46023
[16] Feichtinger, H. G., Generalized amalgams, with applications to Fourier transform, Canad. J. Math., 42, 3, 395-409, (1990) · Zbl 0733.46014
[17] Feichtinger, H. G., Modulation spaces: looking back and ahead, Sampl. Theory Signal Image Process., 5, 2, 109-140, (2006) · Zbl 1156.43300
[18] Feichtinger, H. G.; Gröbner, P., Banach spaces of distributions defined by decomposition methods. I, Math. Nachr., 123, 97-120, (1985) · Zbl 0586.46030
[19] Feichtinger, H. G.; Narimani, G., Fourier multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal., 21, 3, 349-359, (2006) · Zbl 1106.42005
[20] Feichtinger, H. G.; Weisz, F., The Segal algebra \(S_0(\mathbb{R}^d)\) and norm summability of Fourier series and Fourier transforms, Monatsh. Math., 148, 4, 333-349, (2006) · Zbl 1130.42012
[21] Feichtinger, H. G.; Zimmermann, G., Gabor Analysis and Algorithms, A Banach space of test functions for Gabor analysis, 123-170, (1998), Birkhäuser, Boston · Zbl 0890.42008
[22] Fournier, J. J. F.; Stewart, J., Amalgams of \(L^p\) and \(l^q\), Bull. Amer. Math. Soc. (N.S.), 13, 1, 1-21, (1985) · Zbl 0593.43005
[23] Gröchenig, K., Foundations of Time-Frequency Analysis, (2001), Birkhäuser, Boston · Zbl 0966.42020
[24] Heil, C.; Krishna, M.; Radha, R.; Thangavelu, S., Wavelets and Their Applications, An introduction to weighted Wiener amalgams, 183-216, (2003), Allied Publishers, New Delhi
[25] Kobayashi, M.; Sugimoto, M., The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal., 260, 11, 3189-3208, (2011) · Zbl 1232.46033
[26] Lebedev, V.; Olevskiĭ, A., \(C^1\) changes of variable: Beurling-helson type theorem and Hörmander conjecture on Fourier multipliers, Geom. Funct. Anal., 4, 2, 213-235, (1994) · Zbl 0798.42004
[27] Mischler, S.; Mouhot, C., Cooling process for inelastic Boltzmann equations for hard spheres. II. self-similar solutions and tail behavior, J. Stat. Phys., 124, 2-4, 703-746, (2006) · Zbl 1135.82030
[28] Miyachi, A.; Nicola, F.; Rivetti, S.; Tabacco, A.; Tomita, N., Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc., 137, 11, 3869-3883, (2009) · Zbl 1183.42013
[29] Okoudjou, K. A., Embedding of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc., 132, 6, 1639-1647, (2004) · Zbl 1044.46030
[30] Okoudjou, K. A., A Beurling-helson type theorem for modulation spaces, J. Funct. Spaces Appl., 7, 1, 33-41, (2009) · Zbl 1169.42302
[31] Ruzhansky, M.; Sugimoto, M.; Toft, J.; Tomita, N., Changes of variables in modulation and Wiener amalgam spaces, Math. Nachr., 284, 16, 2078-2092, (2011) · Zbl 1228.35279
[32] Ruzhansky, M.; Wang, B.; Zhang, H., Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces, J. Math. Pures Appl. (9), 105, 1, 31-65, (2016) · Zbl 1336.35322
[33] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, 30, (1970), Princeton University Press, Princeton, N.J. · Zbl 0207.13501
[34] Sugimoto, M.; Tomita, N., The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal., 248, 1, 79-106, (2007) · Zbl 1124.42018
[35] Sugimoto, M.; Tomita, N.; Wang, B., Remarks on nonlinear operations on modulation spaces, Integral Transforms Spec. Funct., 22, 4-5, 351-358, (2011) · Zbl 1221.44007
[36] Toft, J., Continuity properties for modulation spaces, with applications to pseudo-differential calculus I, J. Funct. Anal., 207, 2, 399-429, (2004) · Zbl 1083.35148
[37] Wang, B.; Huang, C., Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239, 1, 213-250, (2007) · Zbl 1219.35289
[38] Wang, B.; Lifeng, Z.; Boling, G., Isometric decomposition operators, function spaces \(E_{p, q}^\lambda\) and applications to nonlinear evolution equations, J. Funct. Anal., 233, 1, 1-39, (2006) · Zbl 1099.46023
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