Multilinear embedding – convolution estimates on smooth submanifolds

Beckner, William

doi:10.1090/S0002-9939-2013-11877-0

Multilinear embedding – convolution estimates on smooth submanifolds
HTML articles powered by AMS MathViewer

by William Beckner

Proc. Amer. Math. Soc. 142 (2014), 1217-1228

DOI: https://doi.org/10.1090/S0002-9939-2013-11877-0

Published electronically: December 27, 2013

PDF | Request permission

Abstract:

Multilinear embedding estimates for the fractional Laplacian are obtained in terms of functionals defined over a hyperbolic surface. Convolution estimates used in the proof enlarge the classical framework of the convolution algebra for Riesz potentials to include the critical endpoint index and provide new realizations for fractional integral inequalities that incorporate restriction to smooth submanifolds. Results developed here are modeled on the space-time estimate used by Klainerman and Machedon in their proof of uniqueness for the Gross-Pitaevskii hierarchy.

References

William Beckner, Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1871–1885. MR 2373619, DOI 10.1090/S0002-9939-07-09216-7
William Beckner, Multilinear embedding estimates for the fractional Laplacian, Math. Res. Lett. 19 (2012), no. 1, 175–189. MR 2923184, DOI 10.4310/MRL.2012.v19.n1.a14
Thomas Chen and Nataša Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discrete Contin. Dyn. Syst. 27 (2010), no. 2, 715–739. MR 2600687, DOI 10.3934/dcds.2010.27.715
Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 257819, DOI 10.1007/BF02394567
Kay Kirkpatrick, Benjamin Schlein, and Gigliola Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math. 133 (2011), no. 1, 91–130. MR 2752936, DOI 10.1353/ajm.2011.0004
Sergiu Klainerman and Matei Machedon, Remark on Strichartz-type inequalities, Internat. Math. Res. Notices 5 (1996), 201–220. With appendices by Jean Bourgain and Daniel Tataru. MR 1383755, DOI 10.1155/S1073792896000153
Sergiu Klainerman and Matei Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys. 279 (2008), no. 1, 169–185. MR 2377632, DOI 10.1007/s00220-008-0426-4
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759