## On extremizers for Strichartz estimates for higher order Schrödinger equations

HTML articles powered by AMS MathViewer

- by Diogo Oliveira e Silva and René Quilodrán PDF
- Trans. Amer. Math. Soc.
**370**(2018), 6871-6907 Request permission

## Abstract:

For an appropriate class of convex functions $\phi$, we study the Fourier extension operator on the surface {(𝑦,|𝑦|²+𝜙(𝑦)):𝑦∈ℝ

²}

equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schrödinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations and compute the corresponding operator norms exactly where only lower bounds were previously known.

## References

- Jong-Guk Bak and David McMichael,
*Convolution of a measure with itself and a restriction theorem*, Proc. Amer. Math. Soc.**125**(1997), no. 2, 463–470. MR**1350932**, DOI 10.1090/S0002-9939-97-03569-7 - Matania Ben-Artzi, Herbert Koch, and Jean-Claude Saut,
*Dispersion estimates for fourth order Schrödinger equations*, C. R. Acad. Sci. Paris Sér. I Math.**330**(2000), no. 2, 87–92 (English, with English and French summaries). MR**1745182**, DOI 10.1016/S0764-4442(00)00120-8 - Jonathan Bennett, Neal Bez, Anthony Carbery, and Dirk Hundertmark,
*Heat-flow monotonicity of Strichartz norms*, Anal. PDE**2**(2009), no. 2, 147–158. MR**2547132**, DOI 10.2140/apde.2009.2.147 - Neal Bez and Mitsuru Sugimoto,
*Optimal constants and extremisers for some smoothing estimates*, J. Anal. Math.**131**(2017), 159–187. MR**3631453**, DOI 10.1007/s11854-017-0005-8 - Emanuel Carneiro,
*A sharp inequality for the Strichartz norm*, Int. Math. Res. Not. IMRN**16**(2009), 3127–3145. MR**2533799**, DOI 10.1093/imrn/rnp045 - Emanuel Carneiro and Diogo Oliveira e Silva,
*Some sharp restriction inequalities on the sphere*, Int. Math. Res. Not. IMRN**17**(2015), 8233–8267. MR**3404013**, DOI 10.1093/imrn/rnu194 - Michael Christ and Shuanglin Shao,
*Existence of extremals for a Fourier restriction inequality*, Anal. PDE**5**(2012), no. 2, 261–312. MR**2970708**, DOI 10.2140/apde.2012.5.261 - Michael Christ and Shuanglin Shao,
*On the extremizers of an adjoint Fourier restriction inequality*, Adv. Math.**230**(2012), no. 3, 957–977. MR**2921167**, DOI 10.1016/j.aim.2012.03.020 - Luca Fanelli, Luis Vega, and Nicola Visciglia,
*On the existence of maximizers for a family of restriction theorems*, Bull. Lond. Math. Soc.**43**(2011), no. 4, 811–817. MR**2820166**, DOI 10.1112/blms/bdr014 - Luca Fanelli, Luis Vega, and Nicola Visciglia,
*Existence of maximizers for Sobolev-Strichartz inequalities*, Adv. Math.**229**(2012), no. 3, 1912–1923. MR**2871161**, DOI 10.1016/j.aim.2011.12.012 - Damiano Foschi,
*Maximizers for the Strichartz inequality*, J. Eur. Math. Soc. (JEMS)**9**(2007), no. 4, 739–774. MR**2341830**, DOI 10.4171/JEMS/95 - Damiano Foschi,
*Global maximizers for the sphere adjoint Fourier restriction inequality*, J. Funct. Anal.**268**(2015), no. 3, 690–702. MR**3292351**, DOI 10.1016/j.jfa.2014.10.015 - D. Foschi and D. Oliveira e Silva,
*Some recent progress on sharp Fourier restriction theory*, Anal. Math.**43**(2017), no. 2, 241–265. MR**3685152**, DOI 10.1007/s10476-017-0306-2 - Rupert L. Frank, Elliott H. Lieb, and Julien Sabin,
*Maximizers for the Stein-Tomas inequality*, Geom. Funct. Anal.**26**(2016), no. 4, 1095–1134. MR**3558306**, DOI 10.1007/s00039-016-0380-9 - Wei Han,
*The sharp Strichartz and Sobolev-Strichartz inequalities for the fourth-order Schrödinger equation*, Math. Methods Appl. Sci.**38**(2015), no. 8, 1506–1514. MR**3343568**, DOI 10.1002/mma.3164 - Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal,
*Fundamentals of convex analysis*, Grundlehren Text Editions, Springer-Verlag, Berlin, 2001. Abridged version of*Convex analysis and minimization algorithms. I*[Springer, Berlin, 1993; MR1261420 (95m:90001)] and*II*[ibid.; MR1295240 (95m:90002)]. MR**1865628**, DOI 10.1007/978-3-642-56468-0 - Dirk Hundertmark and Vadim Zharnitsky,
*On sharp Strichartz inequalities in low dimensions*, Int. Math. Res. Not. , posted on (2006), Art. ID 34080, 18. MR**2219206**, DOI 10.1155/IMRN/2006/34080 - Jin-Cheng Jiang, Benoit Pausader, and Shuanglin Shao,
*The linear profile decomposition for the fourth order Schrödinger equation*, J. Differential Equations**249**(2010), no. 10, 2521–2547. MR**2718708**, DOI 10.1016/j.jde.2010.06.014 - Jin-Cheng Jiang, Shuanglin Shao and Betsy Stovall,
*Linear profile decompositions for a family of fourth order Schrödinger equations*, preprint, 2014, arXiv:1410.7520. - Markus Keel and Terence Tao,
*Endpoint Strichartz estimates*, Amer. J. Math.**120**(1998), no. 5, 955–980. MR**1646048**, DOI 10.1353/ajm.1998.0039 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*Oscillatory integrals and regularity of dispersive equations*, Indiana Univ. Math. J.**40**(1991), no. 1, 33–69. MR**1101221**, DOI 10.1512/iumj.1991.40.40003 - P.-L. Lions,
*The concentration-compactness principle in the calculus of variations. The locally compact case. I*, Ann. Inst. H. Poincaré Anal. Non Linéaire**1**(1984), no. 2, 109–145 (English, with French summary). MR**778970**, DOI 10.1016/S0294-1449(16)30428-0 - Diogo Oliveira e Silva,
*Extremizers for Fourier restriction inequalities: convex arcs*, J. Anal. Math.**124**(2014), 337–385. MR**3286057**, DOI 10.1007/s11854-014-0035-4 - Diogo Oliveira e Silva,
*Nonexistence of extremizers for certain convex curves,*preprint, 2012. To appear in Mathematical Research Letters, arXiv:1210.0585. - Benoit Pausader,
*Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case*, Dyn. Partial Differ. Equ.**4**(2007), no. 3, 197–225. MR**2353631**, DOI 10.4310/DPDE.2007.v4.n3.a1 - René Quilodrán,
*Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid*, J. Anal. Math.**125**(2015), 37–70. MR**3317897**, DOI 10.1007/s11854-015-0002-8 - Javier Ramos,
*A refinement of the Strichartz inequality for the wave equation with applications*, Adv. Math.**230**(2012), no. 2, 649–698. MR**2914962**, DOI 10.1016/j.aim.2012.02.020 - Michael Ruzhansky and Mitsuru Sugimoto,
*Smoothing properties of evolution equations via canonical transforms and comparison principle*, Proc. Lond. Math. Soc. (3)**105**(2012), no. 2, 393–423. MR**2959931**, DOI 10.1112/plms/pds006 - 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** - Robert S. Strichartz,
*Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations*, Duke Math. J.**44**(1977), no. 3, 705–714. MR**512086** - Peter A. Tomas,
*A restriction theorem for the Fourier transform*, Bull. Amer. Math. Soc.**81**(1975), 477–478. MR**358216**, DOI 10.1090/S0002-9904-1975-13790-6

## Additional Information

**Diogo Oliveira e Silva**- Affiliation: Hausdorff Center for Mathematics, 53115 Bonn, Germany
- MR Author ID: 756024
- Email: dosilva@math.uni-bonn.de
**René Quilodrán**- Affiliation: Departamento de Ciencias Exactas, Universidad de Los Lagos, Avenida Fuchslocher 1305, Osorno, Chile
- Email: rene.quilodran@ulagos.cl
- Received by editor(s): June 13, 2016
- Received by editor(s) in revised form: December 26, 2016
- Published electronically: February 26, 2018
- Additional Notes: The first author was partially supported by the Hausdorff Center for Mathematics.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 6871-6907 - MSC (2010): Primary 42B10
- DOI: https://doi.org/10.1090/tran/7223
- MathSciNet review: 3841835