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

Sharpness of the Brascamp–Lieb inequality in Lorentz spaces

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.

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, 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.

[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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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.

[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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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.

[1]

Paolo Maremonti. A remark on the Stokes problem in Lorentz spaces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1323-1342. doi: 10.3934/dcdss.2013.6.1323

[2]

Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261

[3]

Manuel Gutiérrez. Lorentz geometry technique in nonimaging optics. Conference Publications, 2003, 2003 (Special) : 386-392. doi: 10.3934/proc.2003.2003.386

[4]

Françoise Pène. Asymptotic of the number of obstacles visited by the planar Lorentz process. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 567-587. doi: 10.3934/dcds.2009.24.567

[5]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665

[6]

Françoise Pène. Self-intersections of trajectories of the Lorentz process. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4781-4806. doi: 10.3934/dcds.2014.34.4781

[7]

A. Carati. On the existence of scattering solutions for the Abraham-Lorentz-Dirac equation . Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 471-480. doi: 10.3934/dcdsb.2006.6.471

[8]

Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043

[9]

Jorge A. Becerril, Javier F. Rosenblueth. Necessity for isoperimetric inequality constraints. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1129-1158. doi: 10.3934/dcds.2017047

[10]

Gisella Croce, Bernard Dacorogna. On a generalized Wirtinger inequality. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1329-1341. doi: 10.3934/dcds.2003.9.1329

[11]

Stéphane Brull, Pierre Degond, Fabrice Deluzet, Alexandre Mouton. Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model. Kinetic & Related Models, 2011, 4 (4) : 991-1023. doi: 10.3934/krm.2011.4.991

[12]

Sun-Sig Byun, Yumi Cho. Lorentz-Morrey regularity for nonlinear elliptic problems with irregular obstacles over Reifenberg flat domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4791-4804. doi: 10.3934/dcds.2015.35.4791

[13]

YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1

[14]

Hubert L. Bray, Marcus A. Khuri. A Jang equation approach to the Penrose inequality. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 741-766. doi: 10.3934/dcds.2010.27.741

[15]

S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 279-291. doi: 10.3934/naco.2012.2.279

[16]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437

[17]

Alexei Shadrin. The Landau--Kolmogorov inequality revisited. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1183-1210. doi: 10.3934/dcds.2014.34.1183

[18]

Igor E. Verbitsky. The Hessian Sobolev inequality and its extensions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6165-6179. doi: 10.3934/dcds.2015.35.6165

[19]

Alain Haraux. Some applications of the Łojasiewicz gradient inequality. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2417-2427. doi: 10.3934/cpaa.2012.11.2417

[20]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143

2016 Impact Factor: 0.483

Metrics

  • PDF downloads (9)
  • HTML views (44)
  • Cited by (0)

[Back to Top]