Estimates for Brascamp-Lieb forms in 𝐿^{𝑝}-spaces with power weights

Brown, R.; Lee, C.; Ott, K.

doi:10.1090/proc/15236

Estimates for Brascamp-Lieb forms in $L^p$-spaces with power weights
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by R. M. Brown, C. W. Lee and K. A. Ott PDF

Proc. Amer. Math. Soc. 149 (2021), 747-760 Request permission

Abstract:

We study a family of Brascamp-Lieb forms acting on families of weighted $L^p$-spaces and Lorentz spaces where the weight is a power of the distance to the origin. We establish a set of necessary conditions and a set of sufficient conditions for the finiteness of these forms in Lorentz spaces. The conditions are close to optimal.

References

Franck Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), no. 2, 335–361. MR 1650312, DOI 10.1007/s002220050267
Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao, Finite bounds for Hölder-Brascamp-Lieb multilinear inequalities, Math. Res. Lett. 17 (2010), no. 4, 647–666. MR 2661170, DOI 10.4310/MRL.2010.v17.n4.a6
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
Neal Bez, Sanghyuk Lee, Shohei Nakamura, and Yoshihiro Sawano, Sharpness of the Brascamp-Lieb inequality in Lorentz spaces, Electron. Res. Announc. Math. Sci. 24 (2017), 53–63. MR 3665018, DOI 10.3934/era.2017.24.006
Herm Jan Brascamp and Elliott H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Advances in Math. 20 (1976), no. 2, 151–173. MR 412366, DOI 10.1016/0001-8708(76)90184-5
R. M. Brown, Estimates for the scattering map associated with a two-dimensional first-order system, J. Nonlinear Sci. 11 (2001), no. 6, 459–471. MR 1871279, DOI 10.1007/s00332-001-0394-8
R. M. Brown, K. A. Ott, and P. A. Perry, Action of a scattering map on weighted Sobolev spaces in the plane, J. Funct. Anal. 271 (2016), no. 1, 85–106. MR 3494243, DOI 10.1016/j.jfa.2016.03.019
E. A. Carlen, E. H. Lieb, and M. Loss, A sharp analog of Young’s inequality on $S^N$ and related entropy inequalities, J. Geom. Anal. 14 (2004), no. 3, 487–520. MR 2077162, DOI 10.1007/BF02922101
Michael Christ, On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc. 287 (1985), no. 1, 223–238. MR 766216, DOI 10.1090/S0002-9947-1985-0766216-6
Loukas Grafakos, On multilinear fractional integrals, Studia Math. 102 (1992), no. 1, 49–56. MR 1164632, DOI 10.4064/sm-102-1-49-56
Loukas Grafakos and Nigel Kalton, Some remarks on multilinear maps and interpolation, Math. Ann. 319 (2001), no. 1, 151–180. MR 1812822, DOI 10.1007/PL00004426
Cong Hoang and Kabe Moen, Weighted estimates for bilinear fractional integral operators and their commutators, Indiana Univ. Math. J. 67 (2018), no. 1, 397–428. MR 3776027, DOI 10.1512/iumj.2018.67.6271
Svante Janson, On interpolation of multilinear operators, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 290–302. MR 942274, DOI 10.1007/BFb0078880
Carlos E. Kenig and Elias M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), no. 1, 1–15. MR 1682725, DOI 10.4310/MRL.1999.v6.n1.a1
Yasuo Komori-Furuya, Weighted estimates for bilinear fractional integral operators: a necessary and sufficient condition for power weights, Collect. Math. 71 (2020), no. 1, 25–37. MR 4047697, DOI 10.1007/s13348-019-00246-5
Kabe Moen, New weighted estimates for bilinear fractional integral operators, Trans. Amer. Math. Soc. 366 (2014), no. 2, 627–646. MR 3130311, DOI 10.1090/S0002-9947-2013-06067-9
Zhongyi Nie and Russell M. Brown, Estimates for a family of multi-linear forms, J. Math. Anal. Appl. 377 (2011), no. 1, 79–87. MR 2754810, DOI 10.1016/j.jmaa.2010.09.070
Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
Peter A. Perry, Global well-posedness and long-time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}(\Bbb {C})$, J. Spectr. Theory 6 (2016), no. 3, 429–481. With an appendix by Michael Christ. MR 3551174, DOI 10.4171/JST/129
E. M. Stein and Guido Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514. MR 0098285, DOI 10.1512/iumj.1958.7.57030
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095