American Institute of Mathematical Sciences

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  2017, 24: 53-63. doi: 10.3934/era.2017.24.006

Sharpness of the Brascamp–Lieb inequality in Lorentz spaces

Neal Bez 1, , Sanghyuk Lee 2, , Shohei Nakamura 3, and  Yoshihiro Sawano 3,

1. 

Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan
2. 

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
3. 

Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan

Received  January 13, 2017 Revised  April 17, 2017 Published  June 2017

Fund Project: This work was supported by JSPS Grant-in-Aid for Young Scientists (A) no. 16H05995 (Bez), NRF (Republic of Korea) Grant no. 2015R1A2A2A05000956 (Lee) and partially supported by JSPS Grand-in-Aid for Scientific Research (C) no. 16K05209 (Sawano). The first author would like to thank Jon Bennett for helpful discussions.

We provide necessary conditions for the refined version of the Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces, thereby establishing the sharpness of the range of Lorentz exponents in the subcritical case. Using similar considerations, some sharp refinements of the Strichartz estimates for the kinetic transport equation are established.

Keywords: Brascamp–Lieb inequality, Lorentz spaces.
Mathematics Subject Classification: 35B45(Primary); 35P10, 35B65(Secondary).
Citation: Neal Bez, Sanghyuk Lee, Shohei Nakamura, Yoshihiro Sawano. Sharpness of the Brascamp–Lieb inequality in Lorentz spaces. Electronic Research Announcements, 2017, 24: 53-63. doi: 10.3934/era.2017.24.006
References:
[1]

K. Astala, D. Faraco and K. Rogers, On Plancherel's identity for a 2D scattering transform, Nonlinearity, 28 (2015), 2721-2729.  doi: 10.1088/0951-7715/28/8/2721.

[2]

K. Ball, Volumes of sections of cubes and related problems, in Geometric Aspects of Functional Analysis (eds. J. Lindenstrauss and V. D. Milman), Springer Lecture Notes in Math., 1376, Springer-Verlag, 1989,251–260. doi: 10.1007/BFb0090058.

[3]

F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math., 134 (1998), 355-361.  doi: 10.1007/s002220050267.

[4]

J. Bennett, N. Bez, T. Flock and S. Lee, Stability of the Brascamp–Lieb constant and applications, to appear in American Journal of Mathematics.

[5]

J. Bennett, N. Bez, S. Gutiérrez and S. Lee, On the Strichartz estimates for the kinetic transport equation, Comm. Partial Differential Equations, 39 (2014), 1821-1826.  doi: 10.1080/03605302.2013.850880.

[6]

J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp-Lieb inequalities: Finiteness, structure and extremals, Geom. Funct. Anal., 17 (2008), 1343-1415.  doi: 10.1007/s00039-007-0619-6.

[7]

J. Bennett, A. Carbery, M. Christ and T. Tao, Finite bounds for Hölder-Brascamp-Lieb multilinear inequalities, Math. Res. Lett., 17 (2010), 647-666.  doi: 10.4310/MRL.2010.v17.n4.a6.

[8]

J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math., 196 (2006), 261-302.  doi: 10.1007/s11511-006-0006-4.

[9]

H. J. Brascamp and E. H. Lieb, Best constants in Young's inequality, its converse, and its generalization to more than three functions, Adv. Math., 20 (1976), 151-173.  doi: 10.1016/0001-8708(76)90184-5.

[10]

R. M. Brown, Estimates for the scattering map associated with a two-dimensional first-order system, J. Nonlinear Sci., 11 (2001), 459-471.  doi: 10.1007/s00332-001-0394-8.

[11]

E. A. Carlen, E. H. Lieb and M. Loss, A sharp analog of Young's inequality on $S^N$ and related entropy inequalities, Jour. Geom. Anal., 14 (2004), 487-520.  doi: 10.1007/BF02922101.

[12]

F. Castella and B. Perthame, Estimations de Strichartz pour les équations de transport cinétique, C. R. Acad. Sci. Paris Sér. I Math., 332 (1996), 535-540. 

[13]

M. Christ, On the restriction of the Fourier transform to curves: Endpoint results and the degenerate case, Trans. Amer. Math. Soc., 287 (1985), 223-238.  doi: 10.1090/S0002-9947-1985-0766216-6.

[14]

G. P. Curbera, J. Garcá-Cuerva, J. María Martell and C. Pérez, Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals, Adv. Math., 203 (2006), 256-318.  doi: 10.1016/j.aim.2005.04.009.

[15]

Z. Guo and L. Peng, Endpoint Strichartz estimate for the kinetic transport equation in one dimension, C. R. Math. Acad. Sci. Paris, 345 (2007), 253-256.  doi: 10.1016/j.crma.2007.07.002.

[16]

L. Guth, The endpoint case in the Bennett-Carbery-Tao multilinear Kakeya conjecture, Acta Math., 205 (2010), 263-286.  doi: 10.1007/s11511-010-0055-6.

[17]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[18]

E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math., 102 (1990), 179-208.  doi: 10.1007/BF01233426.

[19]

Z. Nie and R. M. Brown, Estimates for a family of multi-linear forms, J. Math. Anal. Appl., 377 (2011), 79-87.  doi: 10.1016/j.jmaa.2010.09.070.

[20]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.  doi: 10.1215/S0012-7094-63-03015-1.

[21]

E. Ovcharov, Counterexamples to Strichartz estimates for the kinetic transport equation based on Besicovitch sets, Nonlinear Anal., 74 (2011), 2515-2522.  doi: 10.1016/j.na.2010.12.007.

[22]

E. Ovcharov, Strichartz estimates for the kinetic transport equation, SIAM J. Math. Anal., 43 (2011), 1282-1310.  doi: 10.1137/100803808.

[23]

P. Perry, Global well-posedness and long-time asymptotics for the defocussing Davey-Stewartson Ⅱ equation in $H^{1, 1}(\mathbb{C})$, J. Spectral Theory, 6 (2016), 429-481.  doi: 10.4171/JST/129.

[24]

R. Quilodrán, On extremizing sequences for the adjoint restriction inequality on the cone, J. Lond. Math. Soc., 87 (2013), 223-246.  doi: 10.1112/jlms/jds046.

[25]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, 1971.

[26]

S. I. Valdimarsson, Optimisers for the Brascamp-Lieb inequality, Israel J. Math., 168 (2008), 253-274.  doi: 10.1007/s11856-008-1067-1.

show all references

References:
[1]

K. Astala, D. Faraco and K. Rogers, On Plancherel's identity for a 2D scattering transform, Nonlinearity, 28 (2015), 2721-2729.  doi: 10.1088/0951-7715/28/8/2721.

[2]

K. Ball, Volumes of sections of cubes and related problems, in Geometric Aspects of Functional Analysis (eds. J. Lindenstrauss and V. D. Milman), Springer Lecture Notes in Math., 1376, Springer-Verlag, 1989,251–260. doi: 10.1007/BFb0090058.

[3]

F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math., 134 (1998), 355-361.  doi: 10.1007/s002220050267.

[4]

J. Bennett, N. Bez, T. Flock and S. Lee, Stability of the Brascamp–Lieb constant and applications, to appear in American Journal of Mathematics.

[5]

J. Bennett, N. Bez, S. Gutiérrez and S. Lee, On the Strichartz estimates for the kinetic transport equation, Comm. Partial Differential Equations, 39 (2014), 1821-1826.  doi: 10.1080/03605302.2013.850880.

[6]

J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp-Lieb inequalities: Finiteness, structure and extremals, Geom. Funct. Anal., 17 (2008), 1343-1415.  doi: 10.1007/s00039-007-0619-6.

[7]

J. Bennett, A. Carbery, M. Christ and T. Tao, Finite bounds for Hölder-Brascamp-Lieb multilinear inequalities, Math. Res. Lett., 17 (2010), 647-666.  doi: 10.4310/MRL.2010.v17.n4.a6.

[8]

J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math., 196 (2006), 261-302.  doi: 10.1007/s11511-006-0006-4.

[9]

H. J. Brascamp and E. H. Lieb, Best constants in Young's inequality, its converse, and its generalization to more than three functions, Adv. Math., 20 (1976), 151-173.  doi: 10.1016/0001-8708(76)90184-5.

[10]

R. M. Brown, Estimates for the scattering map associated with a two-dimensional first-order system, J. Nonlinear Sci., 11 (2001), 459-471.  doi: 10.1007/s00332-001-0394-8.

[11]

E. A. Carlen, E. H. Lieb and M. Loss, A sharp analog of Young's inequality on $S^N$ and related entropy inequalities, Jour. Geom. Anal., 14 (2004), 487-520.  doi: 10.1007/BF02922101.

[12]

F. Castella and B. Perthame, Estimations de Strichartz pour les équations de transport cinétique, C. R. Acad. Sci. Paris Sér. I Math., 332 (1996), 535-540. 

[13]

M. Christ, On the restriction of the Fourier transform to curves: Endpoint results and the degenerate case, Trans. Amer. Math. Soc., 287 (1985), 223-238.  doi: 10.1090/S0002-9947-1985-0766216-6.

[14]

G. P. Curbera, J. Garcá-Cuerva, J. María Martell and C. Pérez, Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals, Adv. Math., 203 (2006), 256-318.  doi: 10.1016/j.aim.2005.04.009.

[15]

Z. Guo and L. Peng, Endpoint Strichartz estimate for the kinetic transport equation in one dimension, C. R. Math. Acad. Sci. Paris, 345 (2007), 253-256.  doi: 10.1016/j.crma.2007.07.002.

[16]

L. Guth, The endpoint case in the Bennett-Carbery-Tao multilinear Kakeya conjecture, Acta Math., 205 (2010), 263-286.  doi: 10.1007/s11511-010-0055-6.

[17]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[18]

E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math., 102 (1990), 179-208.  doi: 10.1007/BF01233426.

[19]

Z. Nie and R. M. Brown, Estimates for a family of multi-linear forms, J. Math. Anal. Appl., 377 (2011), 79-87.  doi: 10.1016/j.jmaa.2010.09.070.

[20]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129-142.  doi: 10.1215/S0012-7094-63-03015-1.

[21]

E. Ovcharov, Counterexamples to Strichartz estimates for the kinetic transport equation based on Besicovitch sets, Nonlinear Anal., 74 (2011), 2515-2522.  doi: 10.1016/j.na.2010.12.007.

[22]

E. Ovcharov, Strichartz estimates for the kinetic transport equation, SIAM J. Math. Anal., 43 (2011), 1282-1310.  doi: 10.1137/100803808.

[23]

P. Perry, Global well-posedness and long-time asymptotics for the defocussing Davey-Stewartson Ⅱ equation in $H^{1, 1}(\mathbb{C})$, J. Spectral Theory, 6 (2016), 429-481.  doi: 10.4171/JST/129.

[24]

R. Quilodrán, On extremizing sequences for the adjoint restriction inequality on the cone, J. Lond. Math. Soc., 87 (2013), 223-246.  doi: 10.1112/jlms/jds046.

[25]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, 1971.

[26]

S. I. Valdimarsson, Optimisers for the Brascamp-Lieb inequality, Israel J. Math., 168 (2008), 253-274.  doi: 10.1007/s11856-008-1067-1.

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