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A comparison principle for convolution measures with applications

Part of: Functions of several variables Harmonic analysis in several variables Harmonic analysis in one variable

Published online by Cambridge University Press:  28 June 2019

DIOGO OLIVEIRA E SILVA  and
RENÉ QUILODRÁN
DIOGO OLIVEIRA E SILVA
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT. e-mail: D.OliveiraESilva@bham.ac.uk
RENÉ QUILODRÁN
Affiliation:
e-mail: rquilodr@dim.uchile.cl
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Abstract

We establish the general form of a geometric comparison principle for n-fold convolutions of certain singular measures in ℝd which holds for arbitrary n and d. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [3].

MSC classification

Primary: 42A85: Convolution, factorization
Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 26B10: Implicit function theorems, Jacobians, transformations with several variables
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 307 - 322
Copyright
© Cambridge Philosophical Society 2019

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References

REFERENCES

Bak, J.-G. & McMichael, D.. Convolution of a measure with itself and a restriction theorem. Proc. Amer. Math. Soc. 125 (1997), no. 2, 463–470.CrossRefGoogle Scholar
Bennett, J., Bez, N. & Iliopoulou, M.. Flow monotonicity and Strichartz inequalities. Int. Math. Res. Not. (2015), no. 19, 9415–9437.Google Scholar
Brocchi, G., Oliveira e Silva, D. & Quilodrán, R.. Sharp Strichartz inequalities for fractional and higher order Schrödinger equations. Preprint, 2018. arXiv:1804.11291. To appear in Anal. PDE.Google Scholar
Carneiro, E.. A sharp inequality for the Strichartz norm. Int. Math. Res. Not. (2009), no. 16, 3127–3145.Google Scholar
Carneiro, E., Foschi, D., Oliveira e Silva, D. & Thiele, C.. A sharp trilinear inequality related to Fourier restriction on the circle. Rev. Mat. Iberoam. 33 (2017), no. 4, 1463–1486.CrossRefGoogle Scholar
Carneiro, E. & Oliveira e Silva, D.. Some sharp restriction inequalities on the sphere. Int. Math. Res. Not. (2015), no. 17, 8233–8267.Google Scholar
Carneiro, E., Oliveira e Silva, D. & Sousa, M.. Extremisers for Fourier restriction on hyperboloids. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, 389–415.CrossRefGoogle Scholar
Christ, M. & Shao, S.. Existence of extremals for a Fourier restriction inequality. Anal. PDE 5 (2012), no. 2, 261–312.CrossRefGoogle Scholar
Christ, M. & Shao, S.. On the extremisers of an adjoint Fourier restriction inequality. Adv. Math. 230 (2012), no. 3, 957–977.CrossRefGoogle Scholar
Fefferman, C.. Inequalities for strongly singular convolution operators. Acta Math. 124 (1970), 9–36.CrossRefGoogle Scholar
Foschi, D.. Maximisers for the Strichartz inequality. J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 739–774.CrossRefGoogle Scholar
Foschi, D.. Global maximisers for the sphere adjoint Fourier restriction inequality. J. Funct. Anal. 268 (2015), no. 3, 690–702.CrossRefGoogle Scholar
Foschi, D. & Oliveira e Silva, D.. Some recent progress on sharp Fourier restriction theory. Anal. Math. 43 (2017), no. 2, 241–265.CrossRefGoogle Scholar
Hundertmark, D. & Zharnitsky, V.. On sharp Strichartz inequalities in low dimensions. Int. Math. Res. Not. (2006), Art. ID 34080, 18 pp.CrossRefGoogle Scholar
Oliveira e Silva, D. & Quilodrán, R.. On extremisers for Strichartz estimates for higher order Schrödinger equations. Trans. Amer. Math. Soc. 370 (2018), no. 10, 6871–6907.CrossRefGoogle Scholar
Quilodrán, R.. On extremising sequences for the adjoint restriction inequality on the cone. J. London Math. Soc. 87 (2013), no. 1, 223–246.CrossRefGoogle Scholar
Quilodrán, R.. Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid. J. Anal. Math. 125 (2015), 37–70.CrossRefGoogle Scholar
Ruzhansky, M. & Sugimoto, M.. Smoothing properties of evolution equations via canonical transforms and comparison principle. Proc. Lond. Math. Soc. (3) 105 (2012), no. 2, 393–423.CrossRefGoogle Scholar
Stein, E. M.. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Strichartz, R. S.. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), no. 3, 705–714.CrossRefGoogle Scholar
Tao, T.. Some recent progress on the restriction conjecture. Fourier analysis and convexity, 217–243, Appl. Numer. Harmon. Anal. (Birkhäuser Boston, Boston, MA, 2004).Google Scholar
Tomas, P. A.. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81 (1975), 477–478.CrossRefGoogle Scholar