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Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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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