Optimal constants of smoothing estimates for the 2D and massless 3D Dirac equation

[1] A.A. Balinsky and W.D. Evans, Spectral analysis of relativistic operators,

London Imperial College Press, c2011.

[2] M. Ben-Artzi, Divergence-type operators: spectral theory and spacetime estimates, In: M. Ruzhansky and J. Wirth (Eds.) ”Progress in Mathematics”,

pp. 1-39 (vol. 301) Springer Basel, 2012.

[3] M. Ben-Artzi and A. Devinatz, The limiting absorption principle for partial

differential operators, Mem. Amer. Math. Soc. 66 (1987), no. 364.

[4] M. Ben-Artzi and A. Devinatz, Local smoothing and convergence properties

of Schr¨odinger type equations, J. Funct. Anal. 101 (1991), 231–254.

[5] M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schr¨odinger

equation, J. Anal. Math 58 (1992), 25–37.

[6] M. Ben-Artzi and J. Nemirovsky, Remarks on relativistic Schr¨odinger operators and their extensions, Ann. Inst. H. Poincare-Phys. Theorique 67 (1997),

29-39.

[7] M. Ben-Artzi, M. Ruzhansky and M. Sugimoto, Spectral identities and

smoothing estimates for evolution operators, Adv. Diff. Eqs. 25 (2020), 627650.

[8] M. Ben-Artzi and T. Umeda, Spectral theory of first-order elliptic systems:

from crystal optics to Dirac operators, Rev. Math. Phys. 33 (2021)

[9] N. Bez, H. Saito and M. Sugimoto, Applications of the Funk-Hecke theorem

to smoothing and trace estimates, Adv. Math. 285 (2015), 1767-1795.

[10] N. Bez, M. Sugimoto, Optimal constants and extremisers for some smoothing

estimates, J. Anal. Math. 131 (2017), 159-187.

[11] H. Chihara, Smoothing effects of dispersive pseudodifferential equations,

Comm. Partial Differential Equations 27 (2002), 1953–2005.

[12] P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), 413–439.

[13] P. D’Ancona and L. Fanelli, Strichartz and smoothing for dispersive equations

with magnetic potentials, Comm. Partial Differential Equations 33 (2008),

1082-1112.

23

[14] A. R. Edmonds, Angular momentum in quantum mechanics, Princeton University Press, c1960

[15] T. Hoshiro, Mourre’

s method and smoothing properties of dispersive equations,

Comm. Math. Phys. 202 (1999), 255–265.

[16] T. Hoshiro, Decay and regularity for dispersive equations with constant coefficients, J. Anal. Math. 91 (2003), 211–230.

[17] M. Ikoma, Optimal constants of smoothing estimates for the 2D Dirac equation, J. Fourier Anal. Appl. 28 (2022), Article 57

[18] H. Kalf, T. Okaji and O. Yamada, Explicit uniform bounds on integrals of

Bessel functions and trace theorems for Fourier transforms, Math. Nachr.

292 (2019), 106-120.

[19] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Mathematics, Academic Press, New York (1983),

93–128.

[20] T. Kato, Wave operators and similarity for some non-selfadjoint operators,

Math. Ann. 162 (1965/1966), 258–279.

[21] T. Kato and K. Yajima, Some examples of smooth operators and the associated

smoothing effect, Rev. Math. Phys. 1 (1989), 481-496.

[22] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of

dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69.

[23] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results

for the generalized Korteweg-de Vries equation via the contraction principle,

Comm. Pure Appl. Math. 46 (1993), 527–620.

[24] C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schr¨odinger

equations, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 10 (1993), 255–288.

[25] C. E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc. 342 (1994), 155–172.

[26] C. E. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman

systems, J. Funct. Anal. 127 (1995), 204–234.

[27] C. E. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence

theory for the generalized nonlinear Schr¨odinger equations, Invent. Math. 134

(1998), 489–545.

24

[28] F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. H.

Poincar´e Anal. Non Lin´eaire 10 (1993), 523–548.

[29] C. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc.

Roy. Soc. London Ser. A 306 (1968), 291–296.

[30] M. Ruzhansky and M. Sugimoto, Smoothing properties of evolution equations

via canonical transforms and comparison principle, Proc. London Math. Soc.

(3) 105 (2012), 393–423.

[31] B. Simon, Best constants in some operator smoothness estimates, J. Funct.

Anal. 107 (1992), 66-71.

[32] P. Sj¨olin, Regularity of solutions to the Schr¨odinger equation, Duke Math. J.

55 (1987), 699–715.

[33] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean

Spaces, Princeton Univ. Press, Princeton, N. J., (1971).

[34] M. Sugimoto, A smoothing property of Schr¨odinger equations along the sphere,

J. Anal. Math. 89 (2003), 15–30.

[35] M. Sugimoto, Global smoothing properties of generalized Schr¨olinger equations, J. Anal. Math. 76 (1998), 191-204.

[36] L. Vega, Schr¨odinger equations: pointwise convergence to the initial data,

Proc. Amer. Math. Soc. 102 (1988), 874–878.

[37] M. C. Vilela, Regularity of solutions to the free Schr¨odinger equation with

radial initial data, Illinois J. Math. 45 (2001), 361–370.

[38] B. Walther, A sharp weighted L2 -estimate for the solution to the timedependent Schr¨odinger equation, Ark. Mat. 37 (1989), 381-393.

[39] B. Walther, Regularity, decay, and best constants for dispersive equations, J.

Funct. Anal. 189 (2002) 325–335.

[40] K. Watanabe, Smooth perturbations of the self-adjoint operator |∆|α/2 , Tokyo

J. Math. 14 (1991), 239-250.

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