[1] R. P. Agarwal, N. S. Barnett, S. S. Dragomir, Inequalities for beta and gamma
functions via some classical and new integral inequalities, J. Inequal. Appl. 5 (2000),
103–165.
[2] Y. Aoki, J. Bennett, N. Bez, S. Machihara, K. Matsuura, S. Shiraki, A supersolutions perspective on hypercontractivity, Ann. Mat. Pura Appl. 199 (2020), 2105–
2116.
[3] A. D. Bailey Some results in harmonic analysis related to poinwise convergence,
PhD thesis, University of Birmingham (2012).
[4] D. Bakry, Transformations de Riesz pour les semigroupes sym´etriques, S´eminaire
de Probabilit´es XIX, Lecture Notes in Math., Springer (1985), 145–174.
[5] D. Bakry, Etude
des transformations de Riesz dans les vari´et´es Riemanniennes `
courbure de Ricci minor´ee, S´eminaire de Probabilit´es XXI, Lecture Notes in Math.,
Springer (1987), 137–172.
[6] D. Bakry, F. Bolley, I. Gentil, Dimension dependent hypercontractivity for Gaussian
kernels, Probab. Theory Relat. Fields 154 (2012), 845–874.
[7] D. Bakry, M. Emery,
Diffusions hypercontractives, S´eminaire de Probabilit´es XIX,
Lecture Notes in Math., Springer (1985), 177–206.
[8] D. Bakry, I. Gentil, M. Ledoux, Analysis and geometry of Markov diffusion operators. Grundlehren der mathematischen Wissenschaften 348, Springer (2014).
[9] J. A. Barcel´
o, J. Bennett, A. Carbery, K. M. Rogers, On the dimension of divergence
sets of dispersive equations, Math. Ann. 348 (2011), 599–622.
[10] F. Barthe, The Brunn–Minkowski theorem and related geometric and functional
inequalities, International Congress of Mathematicians. Vol. II, Eur. Math. Soc.,
Z¨
urich (2006), 1529–1546.
[11] M. Beals, Self-spreading and strength of singularities for solutions to semilinear
wave equations, Ann. of Math. 118 (1983), 187–214
[12] W. Beckner, Inequalities in Fourier analysis, Ann. of Math. 102 (1975), 159–182.
[13] W. Beckner, A generalized Poincar´e inequality for Gaussian measures, Proc. Amer.
Math. Soc. 105 (1989), 397–400.
61
[14] D. Beltran, L. Vega, Bilinear identities involving the k-plane transform and Fourier
extension operators, to appear in Proc. Roy. Soc. Edinburgh Sect. A.
[15] J. Bennett, Aspects of multilinear harmonic analysis related to transversality, Harmonic analysis and partial differential equations, Contemp. Math., 612, Amer.
Math. Soc., Providence, RI, (2014), 1–28
[16] J. Bennett, N. Bez, Closure properties of solutions to heat inequalities, J. Geom.
Anal. 19 (2009), 584–600.
[17] J. Bennett, N. Bez, Generating monotone quantities for the heat equation, J. Reine
Angew. Math. 756 (2019), 37–63.
[18] J. Bennett, N. Bez, A. Carbery, D. Hundertmark, Heat-flow monotonicity of
Strichartz norms, Anal. PDE 2 (2009), 147–158.
[19] J. Bennett, N. Bez, M. Iliopoulou, Flow monotonicity and Strichartz inequalities,
Int. Math. Res. Not. IMRN (2015), 9415–9437.
[20] J. Bennett, N. Bez, C. Jeavons, and N. Pattakos, On sharp bilinear Strichartz
estimates of Ozawa–Tsutsumi type, J. Math. Soc. Japan 69 (2017), 459–476.
[21] J. Bennett, N. Bez, T. C. Flock, S. Guti´errez, and M. Iliopoulou, A sharp k-plane
Strichartz inequality for the Schr¨
odinger equation, Trans. Amer. Math. Soc. 370
(2018), 5617–5633.
[22] J. Bennett, A. Carbery, M. Christ, T. Tao, The Brascamp–Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2007), 1343–1415.
[23] J. Bennett, A. Carbery, T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), 261–302.
[24] J. Bennett, A. Carbery, M. Christ, T. Tao, The Brascamp–Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2007), 1343–1415.
[25] J. Bennett, M. Iliopoulou, A multilinear Fourier extension identity on Rd , Math.
Res. Lett. 25 (2018), 1089–1108.
[26] J. Bennett, S. Nakamura, Tomography bounds for the Fourier extension operator
and applications, to appear in Math. Ann.
[27] J. Bennett, K. M. Rogers, On the size of divergence sets for the Schr¨
odinger equation
with radial data, Indiana Univ. Math. J. 61 (2012), 1–13.
[28] N. Bez, K. M. Rogers, A sharp Strichartz estimate for the wave equation with data
in the energy space, J. Eur. Math. Soc. 15 (2013), 805–823.
[29] N. Bez, C. Jeavons, A sharp Sobolev–Strichartz estimate for the wave equation,
Electron. Res. Announc. Math. Sci. 22 (2015), 46–54.
[30] N. Bez, C. Jeavons, T. Ozawa, Some sharp bilinear space-time estimates for the
wave equation, Mathematika 62 (2016), 719–737.
[31] A. Bonami, Etude
des coefficients Fourier des fonctiones de Lp (G), Ann. Inst.
Fourier 20 (1970), 335–402.
62
[32] C. Borell, Positivity improving operators and hypercontractivity, Math. Z. 180
(1982), 225–234.
[33] J. Bourgain, A remark on Schr¨
odinger operators, Israel J. Math. 77 (1992), 1–16.
[34] J. Bourgain, Estimates for cone multipliers, Geometric Aspects of Functional Analysis, Oper. Theory Adv. Appl. 77, Birkh¨auser (1995), 41–60.
[35] J. Bourgain, On the Schr¨
odinger maximal function in higher dimension, Tr. Mat.
Inst. Steklova 280 (2013), 46–60.
[36] J. Bourgain, A note on the Schr¨
odinger maximal function, J. Anal. Math. 130
(2016), 393–396.
[37] J, Bourgain, L. Guth, Bounds on oscillatory integral operators based on multilinear
estimates, Geom. Funct. Anal. 21 (2011), 1239–1295.
[38] E. Carneiro, A sharp inequality for the Strichartz norm, Int. Math. Res. Not. 16
(2009), 3127–3145.
[39] E. Carneiro, D. Oliveira e Silva, M. Sousa, Extremizers for Fourier restriction on
hyperboloids, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 36 (2019), 389–415.
[40] E. Carneiro, D. Oliveira e Silva, M. Sousa, B. Stovall, Extremizers for adjoint
Fourier restriction on hyperboloids: the higher dimensional case, arXiv:1809.05
[41] E. A. Carlen, E. H. Lieb, M. Loss, A sharp analog of Young’s inequality on S N and
related entropy inequalities, J. Geom. Anal. 14 (2004), 487–520.
[42] E. A. Carlen, E. H. Lieb, Optimal hypercontractivity for Fermi fields and related
non-commutative integration inequalities, Commun. Math. Phys. 155 (1993), 27–
46.
[43] L. Carleson, Some analytic problems related to statistical mechanics, In: Euclidean
harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), Lecture
Notes in Math. 779 (1980), 5–45.
[44] C. H. Cho, H. Ko, A note on maximal estimates of generalized Schr¨
odinger equation,
arXiv:1809.03246.
[45] C. H. Cho, Y. Koh, I. Seo, On inhomogeneous Strichartz estimates for fractional
Schr¨
odinger equations and their applications, Discrete Contin. Dyn. Syst. 36 (2016),
1905–1926.
[46] C. H. Cho, S. Lee, Dimension of divergence sets for the pointwise convergence of
the Schr¨
odinger equation, J. Math. Anal. Appl. 411 (2014), 254–260.
[47] Y. Cho, S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math.
J. 62 (2013), 991–1020.
[48] C. H. Cho, S. Lee, A. Vargas, Problems on pointwise convergence of solutions to
the Schr¨
odinger equation, J. Fourier Anal. Appl. 18 (2012), 972–994.
[49] C. H. Cho, S. Shiraki, Pointwise convergence along a tangential curve for the fractional Schr¨
odinger equation, to appear in Ann. Acad. Sci. Fenn. Math.
63
[50] P. Constantin, J.-C. Saut, Local smoothing properties of dispersive equations, J.
Amer. Math. Soc. 1 (1988), 413–439.
[51] M. G. Cowling, Pointwise behavior of solutions to Schr¨
odinger equations, Harmonic
Analysis (Cortona, 1982), Lecture Notes in Math. 992 (1983), 83–90.
[52] B. E. J. Dahlberg, C. E. Kenig, A note on the almost everywhere behavior of solutions to the Schr¨
odinger equation. In: Harmonic analysis (Minneapolis, Minn.,
1981), Lecture Notes in Math. 908 (1982), 205–209.
[53] E. B. Davies, L. Gross, B. Simon. Hypercontractivity: a bibliographic review, Ideas
and methods in quantum and statistical physics (Oslo, 1988)(1992), 370–389.
[54] C. Demeter, S. Guo, Schr¨
odinger maximal function estimates via the pseudoconformal transformation, arXiv:1608.07640v1.
[55] Y. Ding, Y. Niu, Convergence of solutions of general dispersive equations along
curve, Chin. Ann. Math. Ser. B 40 (2019), 363–388.
[56] X. Du, L. Guth, X. Li, A sharp Schr¨
odinger maximal estimate in R2 , Ann. of Math.
186 (2017), 607–640.
[57] X. Du, R. Zhang, Sharp L2 estimate of Schr¨
odinger maximal function in higher
dimensions, Ann. of Math. 189 (2019), 837–861.
[58] X. Du, L. Guth, X. Li, R. Zhang, Pointwise convergence of Schr¨
odinger solutions
and multilinear refined Strichartz estimates Forum Math. Sigma 6 (2018), e14, 18
pp.
[59] P. Federbush, Partially Alternate Derivation of a Result of Nelson, Journal of Mathematical Physics 10 (1969), 50–52.
[60] D. Foschi, On an endpoint case of the Klainerman–Machedon estimates, Comm.
Partial Differential Equations 25 (2000), 1537–1547.
[61] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. 9 (2007),
739–774.
[62] D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave
equations, Ann. Sci. Ecole Norm. Sup. 33 (2000), 211–274.
[63] D. Foschi, D. Oliveira e Silva, Some recent progress in sharp Fourier restriction
theory, Anal. Math. 43 (2017), 241–265
[64] J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation, Partial differential operators and mathematical physics (Holzhau, 1994), Oper. Theory
Adv. Appl., 78, Birkh¨
auser, Basel (1995), 153–160.
[65] J. Glimm, Boson fields with nonlinear selfinteraction in two dimensions, Commun.
math. Phys. 8 (1968), 12–25.
[66] L. Gross, Existence and uniqueness of physical ground states, J. Functional Analysis
10 (1972), 52–109.
[67] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061–1083.
64
[68] L. Gross, Hypercontractivity, logarithmic Sobolev inequalities, and applications: a
survey of surveys, Diffusion, quantum theory, and radically elementary mathematics, Math. Notes 47, Princeton Univ. Press (2006), 45–73.
[69] F. Guerra, Euclidean Field Theory, arXiv:math-ph/0510087.
[70] F. Guerra, L. Rosen, B. Simon, Nelson’s symmetry and the infinite volume behavior
of the vacuum in P (φ)2 , Comm. Math. Phys. 27 (1972), 10–22.
[71] B. Guo, Z. Huo, Global well-posedness for the fractional nonlinear Schr¨
odinger
equation, Comm. Partial Differential Equations 36 (2011), 247–255.
[72] Z. Guo, L. Li, K. Nakanishi, L. Yan, On the boundary Strichartz estimates for wave
and Schr¨
odinger equations, J. Differential Equations 265 (2018), 5656–5675.
[73] Z. Guo, Y. Wang, Improved Strichartz estimates for a class of dispersive equations in
the radial case and their applications to nonlinear Schr¨
odinger and wave equations,
J. Anal. Math. 124 (2014), 1–38.
[74] L. Guth, The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture, Acta Math. 205 (2010), 263–286.
[75] L. Guth, A short proof of the multilinear Kakeya inequality, Math. Proc. Cambridge
Philos. Soc. 158 (2015), 147–153.
[76] E. Gwynne Functional Inequalities for Gaussian and Log-Concave Probability Measures, PhD thesis, Northwestern University (2013).
[77] E. Gwynne, E. P. Hsu, On Beckner’s inequality for Gaussian measures, Elem. Math.
70 (2015), 33–38.
[78] R. Høegh-Krohn, B. Simon, Hypercontractive semigroups and two dimensional selfcoupled Bose fields, J. Functional Analysis 9 (1972), 121–180.
[79] Y. Hong, Y. Sire, On fractional Schr¨
odinger equations in Sobolev spaces, Commun.
Pure Appl. Anal. 14 (2015), 2265–2282.
[80] Y. Hu, A unified approach to several inequalities for Gaussian and diffusion measures, S´eminaire de Probabilit´es XXXIV. Lecture Notes in Math. 1729, Springer
(2000), 329–335.
[81] Y. Hu, Analysis on Gaussian spaces, World Scientific Publishing Co. (2017).
[82] D. Hundertmark, V. Zharnitsky, On sharp Strichartz inequalities in low dimensions,
Int. Math. Res. Not. (2006), Art. ID 34080, 18 pp.
[83] A. D. Ionescu, F. Pusateri, Nonlinear fractional Schr¨
odinger equations in one dimension, J. Funct. Anal. 266 (2014), 139–176.
[84] C. Jeavons, A sharp bilinear estimate for the Klein–Gordon equation in arbitrary
space-time dimensions, Differential Integral Equations 27 (2014), 137–156.
[85] C. Jeavons, Optimal constants and maximising functions for Strichartz inequalities,
PhD thesis, University of Birmingham (2015).
65
[86] Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl.
387 (2012), 857–861.
[87] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955–
980.
[88] C. E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive
equations, Indiana Univ. Math. J. 40 (1991), 33–69.
[89] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local
existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221–1268.
[90] S. Klainerman and M. Machedon, Remark on Strichartz-type inequalities. With
appendices by Jean Bourgain and Daniel Tataru. Internat. Math. Res. Notices 5
(1996), 201–220.
[91] S. Klainerman and M. Machedon, On the regularity properties of a model problem
related to wave maps, Duke Math. J. 87 (1997), 553–589
[92] S. Klainerman and S. Selberg, Remark on the optimal regularity for equations of
wave maps type, Comm. Part. Dif. Eq. 22 (1997), 901–918.
[93] S. Klainerman and D. Tataru, On the optimal regularity for the Yang–Mills equations in R4+1 , J. Amer. Math. Soc. 12 (1999), 93–116.
[94] M. Kunze, On the existence of a maximizer for the Strichartz inequality, Comm.
Math. Phys. 243 (2003), 137–162.
[95] N. Laskin, Fractional quantum mechanics and L´evy path integrals, Phys. Lett. A
268 (2000), 298–305.
[96] N. Laskin, Fractional Schr¨
odinger equation, Phys. Rev. E 66 (2002), 056108.
[97] S. Lee, On pointwise convergence of the solutions to Schr¨
odinger equations in R2 ,
Int. Math. Res. Not. (2006), 1–21.
[98] S. Lee, Bilinear restriction estimates for surfaces with curvatures of different signs,
Trans. Amer. Math. Soc. 358 (2006), 3511–3533.
[99] S. Lee, K. Rogers, A. Vargas, Sharp null form estimates for the wave equation in
R3+1 , Int. Math. Res. Not. IMRN 2008, Art. ID rnn 096, 18 pp.
[100] S. Lee, K. Rogers, The Schr¨
odinger equation along curves and the quantum harmonic oscillator, Adv. Math. 229 (2012), 1359–1379.
[101] S. Lee, A. Vargas, Sharp null form estimates for the wave equation, Amer. J. Math.
130 (2008), 1279–1326.
[102] M. Ledoux, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse
Math. IX (2000), 305–366.
[103] M. Ledoux, Remarks on Gaussian noise stability, Brascamp–Lieb and Slepian inequalities, Geometric aspects of functional analysis, Lecture Notes in Math. 2116,
Springer (2014), 309–333.
66
[104] M. Ledoux, Heat flows, geometric and functional inequalities, Proceedings of the
International Congress of Mathematicians, Seoul (2014), Vol. IV, 117–135.
[105] R. Luc`
a, K. M. Rogers, An improved necessary condition for the Schr¨
odinger maximal estimate, arXiv:1506.05325.
[106] R. Luc`
a, K. M. Rogers, Coherence on fractals versus pointwise convergence for the
Schr¨
odinger equation, Comm. Math. Phys. 351 (2017), 341–359.
[107] R. Luc`
a, K. M. Rogers, A note on pointwise convergence for the Schr¨
odinger
equation, Math. Proc. Cambridge Philos. Soc. 166 (2019), 209–218.
[108] R. Luc`
a, K. M. Rogers, Average decay of the Fourier transform of measures with
applications, J. Eur. Math. Soc. (JEMS) 21 (2019), 465–506.
[109] J. Matkowski, The converse of the H¨
older inequality and its generalizations, Studia
Math. 109 (1994), 171–182.
[110] P. Mattila, Geometry of Sets and Measures in Eucldean Spaces, Fractals and Rectifiability, Cambridge Univ. Press, Cambridge (1995).
[111] A. Montanaro, Some applications of hypercontractive inequalities in quantum information theory, J. Math. Phys. 53 (2012), 122206.
[112] E. Mossel, R. O’Donnell, O. Regev, J. E. Steif, B. Sudakov, Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami–Beckner
inequality, Israel J. Math. 154 (2006), 299–336.
[113] E. Mossel , K. Oleszkiewicz, A. Sen, On reverse hypercontractivity, Geom. Funct.
Anal. 23, (2013), 1062–1097.
[114] A. Moyua, A. Vargas, L. Vega, Schr¨
odinger maximal function and restriction properties of the Fourier transform, Internat. Math. Res. Notices 1996, 793–815.
[115] A. Moyua, A. Vargas, L. Vega, Restriction theorems and maximal operators related
to oscillatory integrals in R3 , Duke Math. J. 96 (1999), 547–574.
[116] K. Nakanishi, Energy scattering for nonlinear Klein–Gordon and Schr¨
odinger
equations in spatial dimensions 1 and 2, J. Funct. Anal. 169 (1999), 201–225.
[117] E. Nelson, A quartic interaction in two dimensions, in Mathematical Theory of
Elementary Particles, (R. Goodman and I. Segal. eds. ) M.I.T. Press. Cambridge.
fass. (1966), 69–73.
[118] E. Nelson, Construction of quantum fields from Markov fields, J. Funct. Anal. 12
(1973), 97–112.
[119] E. Nelson, The free Markov field, J. Funct. Anal. 12 (1973), 211–227.
[120] B. Pausader, The cubic fourth-order Schr¨
odinger equation, J. Funct. Anal. 256
(2009), 2473–2517.
[121] K. M. Rogers, P. Villarroya, Sharp estimates for maximal operators associated to
the wave equation, Ark. Mat. 46 (2008),143–151.
67
[122] R. O’Donnell, Social choice, computational complexity, Gaussian geometry, and
Boolean functions, Proceedings of the International Congress of Mathematicians
2014, Volume IV, 633–658.
[123] T. Ozawa, K. Rogers, A sharp bilinear estimate for the Klein–Gordon equation in
R1+1 , Int. Math. Res. Not. (2014) 1367–1378.
[124] T. Ozawa, Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear
Schr¨
odinger equations, Differential Integral Equations 11 (1998), 201–222
[125] F. Planchon, L. Vega, Bilinear virial identities and applications, Ann. Sci. Ec.
Norm. Sup´er. 42 (2009), 261–290.
[126] R. Quilodr´
an, Nonexistence of extremals for the adjoint restriction inequality on
the hyperboloid, J. Anal. Math. 125 (2015), 37–70.
[127] I. E. Segal, Tensor algebras over Hilbert spaces, Trans. Amer. Math. Sot. 81 (1956),
106–134.
[128] S. Selberg, Almost optimal local well-posedness of the Maxwell–Klein–Gordon equations in 1 + 4 dimensions, Commun. PDE 27 (2002), 1183–1227.
[129] S. Selberg, A. Tesfahun, Finite-energy global well-posedness of the Maxwell–Klein–
Gordon system in Lorenz gauge, Commun. Partial Differ. Equ. 35 (2010), 1029–
1057.
[130] B. Simon, Best constants in some operator smoothness estimates, J. Funct. Anal.
107 (1992), 66–71.
[131] B. Simon, Chapter Three. Ed Nelson’s Work in Quantum Theory, In Diffusion,
Quantum Theory, and Radically Elementary Mathematics (2006), 75–94.
[132] S. Shiraki, Pointwise convergence along restricted directions for the fractional
Schr¨
odinger equation, J. Fourier Anal. Appl. 26 (2020), 12 pp.
[133] P. Sj¨
ogren, P. Sj¨
olin, Convergence properties for the time dependent Schr¨
odinger
equation, Ann. Acad. Sci. Fenn. A I Math. 14 (1989), 13–25.
[134] P. Sj¨
olin, Regularity of solutions to the Schr¨
odinger equation, Duke Math. J. 55
(1987), 699–715.
[135] E. M. Stein, Harmonic Analysis, Princeton Univ. Press, Princeton (1993).
[136] T. Tao, Endpoint bilinear restriction theorems for the cone, and some sharp null
form estimates, Math. Z. 238 (2001), 215–268.
[137] T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal.
13 (2003), 1359–1384.
[138] T. Tao, Sharp bounds for multilinear curved Kakeya, restriction and oscillatory
integral estimates away from the endpoint, Mathematika 66 (2020), 517–576.
[139] T. Tao, A. Vargas, A bilinear approach to cone multipliers. I. Restriction estimates,
Geom. Funct. Anal. 10 (2000), 185–215.
68
[140] T. Tao, A. Vargas, A bilinear approach to cone multipliers. II. Applications, Geom.
Funct. Anal. 10 (2000), 216–258.
[141] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya
conjectures, J. Amer. Math. Soc. 11 (1998), 967–1000.
[142] D. Tataru, Null form estimates for second order hyperbolic operators with rough
coefficients, Harmonic analysis at Mount Holyoke, 383–409.
[143] L. Vega, Schr¨
odinger equations: pointwise convergence to the initial data, Proc.
Amer. Math. Soc. 102 (1988), 874–878.
[144] B. G. Walther, Maximal estimates for oscillatory integrals with concave phase,
Contemp. Math. 189 (1995), 485–495.
[145] B. G. Walther, Some Lp (L∞ )-and L2 (L2 )-estimates for oscillatory Fourier transforms, Analysis of divergence (Orono, ME, 1997), 213–231.
[146] T. Wolff, A sharp bilinear cone restriction estimate, Ann. of Math. 153 (2001),
661–698.
[147] D. Zubrini´
c, Singular sets of Sobolev functions, C. R. Math. Acad. Sci. Paris 334
(2002), 539–544.
69
...
論文の公開元へ
