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Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres. (English) Zbl 1458.35011

Summary: We prove that constant functions are the unique real-valued maximizers for all \(L^2 - L^{2 n}\) adjoint Fourier restriction inequalities on the unit sphere \(\mathbb{S}^{d - 1} \subset \mathbb{R}^d, d \in \{3, 4, 5, 6, 7\}\), where \(n \geqslant 3\) is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler-Lagrange equation being smooth, a fact of independent interest which we establish in the companion paper [the authors, Forum Math. Sigma 9, Paper No. e12, 40 p. (2021; Zbl 1467.45006)]. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character \(e^{i \xi \cdot \omega}\), for some \(\xi\), thereby extending previous work of M. Christ and S. Shao [Adv. Math. 230, No. 3, 957–977 (2012; Zbl 1258.35007)] to arbitrary dimensions \(d \geqslant 2\) and general even exponents.

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B37 Harmonic analysis and PDEs
45C05 Eigenvalue problems for integral equations
51M16 Inequalities and extremum problems in real or complex geometry
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[1] Abi-Khuzam, F., Inequalities and asymptotics for some moment integrals, J. Inequal. Appl., Article 257 pp. (2017), 8 pp. · Zbl 1375.26035
[2] Antoneli, F.; Forger, M.; Gaviria, P., Maximal subgroups of compact Lie groups, J. Lie Theory, 22, 4, 949-1024 (2012) · Zbl 1261.22007
[3] Ball, K., Cube slicing in \(\mathbb{R}^n\), Proc. Am. Math. Soc., 97, 3, 465-473 (1986) · Zbl 0601.52005
[4] Borwein, J.; Chan, O-Y., Uniform bounds for the complementary incomplete gamma function, Math. Inequal. Appl., 12, 1, 115-121 (2009) · Zbl 1177.33009
[5] Borwein, J.; Sinnamon, C., A closed form for the density functions of random walks in odd dimensions, Bull. Aust. Math. Soc., 93, 2, 330-339 (2016) · Zbl 1347.60049
[6] Borwein, J.; Straub, A.; Vignat, C., Densities of short uniform random walks in higher dimensions, J. Math. Anal. Appl., 437, 1, 668-707 (2016) · Zbl 1333.60086
[7] Borwein, J.; Straub, A.; Wan, J.; Zudilin, W., Densities of short uniform random walks, Can. J. Math., 64, 5, 961-990 (2012), With an appendix by Don Zagier · Zbl 1296.33011
[8] Brislawn, C., Kernels of trace class operators, Proc. Am. Math. Soc., 104, 4, 1181-1190 (1988) · Zbl 0695.47017
[9] Brislawn, C., Traceable integral kernels on countably generated measure spaces, Pac. J. Math., 150, 2, 229-240 (1991) · Zbl 0724.47014
[10] Brocchi, G.; Oliveira e. Silva, D.; Quilodrán, R., Sharp Strichartz inequalities for fractional and higher order Schrödinger equations, Anal. PDE, 13, 2, 477-526 (2020) · Zbl 1435.35015
[11] Brzezinski, P., Schnittvolumina hochdimensionaler konvexer Körper (2011), Christian-Albrechts-Universität zu Kiel, PhD thesis
[12] Brzezinski, P., Volume estimates for sections of certain convex bodies, Math. Nachr., 286, 17-18, 1726-1743 (2013) · Zbl 1285.52003
[13] Carneiro, E.; Oliveira e. Silva, D., Some sharp restriction inequalities on the sphere, Int. Math. Res. Not., 17, 8233-8267 (2015) · Zbl 1325.42008
[14] 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, 4, 1463-1486 (2017) · Zbl 1390.42012
[15] Carneiro, E.; Oliveira e. Silva, D.; Sousa, M., Sharp mixed norm spherical restriction, Adv. Math., 341, 583-608 (2019) · Zbl 1412.42030
[16] Christ, M., Extremizers of a Radon transform inequality, (Advances in Analysis: the Legacy of Elias M. Stein. Advances in Analysis: the Legacy of Elias M. Stein, Princeton Math. Ser., vol. 50 (2014), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ), 84-107 · Zbl 1303.44001
[17] Christ, M.; Shao, S., Existence of extremals for a Fourier restriction inequality, Anal. PDE, 5, 2, 261-312 (2012) · Zbl 1273.42009
[18] Christ, M.; Shao, S., On the extremizers of an adjoint Fourier restriction inequality, Adv. Math., 230, 3, 957-977 (2012) · Zbl 1258.35007
[19] deLyra, J. L., On the sums of inverse even powers of zeros of regular Bessel functions, Preprint at
[20] Dirksen, H., Sections of simplices and cylinders, volume formulas and estimates (2015), Christian-Albrechts-Universität zu Kiel, PhD dissertation
[21] Dirksen, H., Hyperplane sections of cylinders, Colloq. Math., 147, 1, 145-164 (2017) · Zbl 1368.52005
[22] Fanelli, L.; Vega, L.; Visciglia, N., On the existence of maximizers for a family of restriction theorems, Bull. Lond. Math. Soc., 43, 4, 811-817 (2011) · Zbl 1225.42012
[23] Foschi, D., Global maximizers for the sphere adjoint Fourier restriction inequality, J. Funct. Anal., 268, 690-702 (2015) · Zbl 1311.42019
[24] Foschi, D.; Oliveira e. Silva, D., Some recent progress on sharp Fourier restriction theory, Anal. Math., 43, 2, 241-265 (2017) · Zbl 1389.42016
[25] Frank, R.; Lieb, E. H.; Sabin, J., Maximizers for the Stein-Tomas inequality, Geom. Funct. Anal., 26, 4, 1095-1134 (2016) · Zbl 1357.42023
[26] García-Pelayo, R., Exact solutions for isotropic random flights in odd dimensions, J. Math. Phys., 53, Article 103504 pp. (2012) · Zbl 1290.82012
[27] Garsia, A.; Rodemich, E.; Rumsey, H., On some extremal positive definite functions, J. Math. Mech., 18, 805-834 (1968/1969) · Zbl 0191.06002
[28] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (2007), Elsevier/Academic Press: Elsevier/Academic Press Amsterdam, Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger · Zbl 1208.65001
[29] Greenwood, J. A.; Durand, D., The distribution of length and components of the sum of n random unit vectors, Ann. Math. Stat., 26, 233-246 (1955) · Zbl 0066.12401
[30] Hall, B., Lie Groups, Lie Algebras, and Representations. An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2015), Springer-Verlag: Springer-Verlag New York · Zbl 1316.22001
[31] Hofmann, K. H., Lie algebras with subalgebras of co-dimension one, Ill. J. Math., 9, 636-643 (1965) · Zbl 0142.27601
[32] Ifantis, E. K.; Siafarikas, P. D., Inequalities involving Bessel and modified Bessel functions, J. Math. Anal. Appl., 147, 1, 214-227 (1990) · Zbl 0709.33003
[33] Jurdjevic, V.; Sussmann, H. J., Control systems on Lie groups, J. Differ. Equ., 12, 313-329 (1972) · Zbl 0237.93027
[34] Kerman, R.; Ol’hava, R.; Spektor, S., An asymptotically sharp form of Ball’s integral inequality, Proc. Am. Math. Soc., 143, 9, 3839-3846 (2015) · Zbl 1408.33003
[35] Kershaw, D., Some extensions of W. Gautschi’s inequalities for the gamma function, Math. Comput., 41, 164, 607-611 (1983) · Zbl 0536.33002
[36] König, H., On the best constants in the Khintchine inequality for Steinhaus variables, Isr. J. Math., 203, 1, 23-57 (2014) · Zbl 1314.46017
[37] König, H.; Koldobsky, A., On the maximal measure of sections of the n-cube, (Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations. Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, Contemp. Math., vol. 599 (2013), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 123-155 · Zbl 1318.52003
[38] König, H.; Koldobsky, A., On the maximal perimeter of sections of the cube, Adv. Math., 346, 773-804 (2019) · Zbl 1410.52006
[39] König, H.; Kwapień, S., Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors, Positivity, 5, 2, 115-152 (2001) · Zbl 0998.60018
[40] Krasikov, I., Approximations for the Bessel and Airy functions with an explicit error term, LMS J. Comput. Math., 17, 1, 209-225 (2014) · Zbl 1294.41024
[41] Krein, M. G.; Rutman, M. A., Linear operators leaving invariant a cone in a Banach space, Am. Math. Soc. Transl., 1950, 26 (1950), 128 pp.
[42] Laforgia, A., Further inequalities for the gamma function, Math. Comput., 42, 166, 597-600 (1984) · Zbl 0536.33003
[43] Laforgia, A.; Muldoon, M. E., Inequalities and approximations for zeros of Bessel functions of small order, SIAM J. Math. Anal., 14, 2, 383-388 (1983) · Zbl 0514.33006
[44] Landau, L. J., Bessel functions: monotonicity and bounds, J. Lond. Math. Soc. (2), 61, 1, 197-215 (2000) · Zbl 0948.33001
[45] Mann, L. N., Gaps in the dimensions of transformation groups, Ill. J. Math., 10, 532-546 (1966) · Zbl 0142.21701
[46] Mordhorst, O., The optimal constants in Khintchine’s inequality for the case \(2 < p < 3\), Colloq. Math., 147, 2, 203-216 (2017) · Zbl 1370.26047
[47] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.23 of 2019-06-15. F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders (Eds.).
[48] Natalini, P.; Palumbo, B., Inequalities for the incomplete gamma function, Math. Inequal. Appl., 3, 1, 69-77 (2000) · Zbl 0979.33001
[49] Nazarov, F. L.; Podkorytov, A. N., Ball, Haagerup, and distribution functions, (Complex Analysis, Operators, and Related Topics. Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl., vol. 113 (2000), Birkhäuser: Birkhäuser Basel), 247-267 · Zbl 0969.46014
[50] Oleszkiewicz, K.; Pełczyński, A., Polydisc slicing in \(\mathbb{C}^n\), Stud. Math., 142, 3, 281-294 (2000) · Zbl 0971.32018
[51] Oliveira e. Silva, D.; Quilodrán, R., Smoothness of solutions of a restricted convolution equation on spheres (2019), Preprint at
[52] Oliveira e. Silva, D.; Thiele, C., Estimates for certain integrals of products of six Bessel functions, Rev. Mat. Iberoam., 33, 4, 1423-1462 (2017) · Zbl 1384.33014
[53] Oliveira e. Silva, D.; Thiele, C.; Zorin-Kranich, P., Band-limited maximizers for a Fourier extension inequality on the circle, Exp. Math. (2019)
[54] Pearson, K., Mathematical contributions to the theory of evolution. IX. A mathematical theory of random migration, (Drapers’ Company Research Memoirs. Drapers’ Company Research Memoirs, Biometric Series, vol. 3 (1906))
[55] Piessens, R.; de Doncker-Kapenga, E.; Uberhuber, C. W.; Kahaner, D. K., QUADPACK: A Subroutine Package for Automatic Integration (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0508.65005
[56] Sawa, J., The best constant in the Khintchine inequality for complex Steinhaus variables, the case \(p = 1\), Stud. Math., 81, 1, 107-126 (1985) · Zbl 0612.60017
[57] Shao, S., On existence of extremizers for the Tomas-Stein inequality for \(\mathbb{S}^1\), J. Funct. Anal., 270, 3996-4038 (2016) · Zbl 1339.42011
[58] Sneddon, I. N., On some infinite series involving the zeros of Bessel functions of the first kind, Proc. Glasgow Math. Assoc., 4, 144-156 (1960) · Zbl 0094.04301
[59] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (1993), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0821.42001
[60] Steinhaus, H., Sur les distances des points dans les ensembles de mesure positive, Fundam. Math., 1, 93-104 (1920) · JFM 47.0179.02
[61] Stromberg, K., An elementary proof of Steinhaus’s theorem, Proc. Am. Math. Soc., 36, 308 (1972) · Zbl 0278.28013
[62] Tomas, P., A restriction theorem for the Fourier transform, Bull. Am. Math. Soc., 81, 2, 477-478 (1975) · Zbl 0298.42011
[63] Van Deun, J.; Cools, R., Algorithm 858: computing infinite range integrals of an arbitrary product of Bessel functions, ACM Trans. Math. Softw., 32, 4, 580-596 (2006), Associated computer program available online · Zbl 1230.65027
[64] Van Deun, J.; Cools, R., A Matlab implementation of an algorithm for computing integrals of products of Bessel functions, (Mathematical Software. Mathematical Software, ICMS 2006. Mathematical Software. Mathematical Software, ICMS 2006, Lecture Notes in Comput. Sci., vol. 4151 (2006), Springer: Springer Berlin), 284-295 · Zbl 1230.33015
[65] Van Deun, J.; Cools, R., Integrating products of Bessel functions with an additional exponential or rational factor, Comput. Phys. Commun., 178, 8, 578-590 (2008), Associated computer program available online · Zbl 1196.65059
[66] Watson, G. N., A Treatise on the Theory of Bessel Functions (1944), Cambridge University Press/The Macmillan Company: Cambridge University Press/The Macmillan Company Cambridge, England/New York · Zbl 0063.08184
[67] Werner, D., Funktionalanalysis (2018), Springer-Verlag: Springer-Verlag Berlin · Zbl 1395.46001
[68] Zhou, Y., Wick rotations, Eichler integrals, and multi-loop Feynman Diagrams, Commun. Number Theory Phys., 12, 1, 127-192 (2018) · Zbl 1393.81029
[69] Zhou, Y., On Borwein’s conjectures for planar uniform random walks, J. Aust. Math. Soc., 107, 3, 392-411 (2019) · Zbl 1472.60081
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