Estimates of one-dimensional oscillatory integrals
Annales de l'Institut Fourier, Volume 33 (1983) no. 4, p. 189-201
We study one-dimensional oscillator integrals which arise as Fourier-Stieltjes transforms of smooth, compactly supported measures on smooth curves in Euclidean spaces and determine their decay at infinity, provided the curves satisfy certain geometric conditions.
Nous considérons des intégrales oscillatoires, de dimension un, qui sont transformées de Fourier-Stieltjes de mesures suffisamment régulières à support compact sur des courbes indéfiniment dérivables dans des espaces euclidiens. Nous déterminons leur comportement à l’infini pourvu qu’ils satisfassent certaines conditions géométriques.
@article{AIF_1983__33_4_189_0,
     author = {Muller, Detlef},
     title = {Estimates of one-dimensional oscillatory integrals},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {33},
     number = {4},
     year = {1983},
     pages = {189-201},
     doi = {10.5802/aif.945},
     zbl = {0511.42013},
     mrnumber = {86f:42003},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1983__33_4_189_0}
}
Muller, Detlef. Estimates of one-dimensional oscillatory integrals. Annales de l'Institut Fourier, Volume 33 (1983) no. 4, pp. 189-201. doi : 10.5802/aif.945. http://www.numdam.org/item/AIF_1983__33_4_189_0/

[1] F. Carlson, Une inégalité, Ark. Mat. Astr. Fys., 25, B1 (1934). | JFM 61.0206.02 | Zbl 0009.34202

[2] L. Corwin, F. P. Greenleaf, Singular Fourier integral operators and representations of nilpotent Lie groups, Comm. on Pure and Applied Math., B1 (1978), 681-705. | MR 81f:46055 | Zbl 0391.46033

[3] Y. Domar, On the Banach algebra A(Γ) for smooth sets Γ ⊂Rn, Comment. Math. Helv., 52 (1977), 357-371. | MR 57 #17121 | Zbl 0356.43002

[4] C. S. Herz, Fourier transforms related to convex sets, Ann. of Math., (2), 75 (1962), 215-254. | MR 26 #545 | Zbl 0111.34803

[5] L. Hörmander, Lower bounds at infinity for solutions of differential equations with constant coefficients, Israel J. Math., 16 (1973), 103-116. | MR 49 #5543 | Zbl 0271.35005

[6] W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull, Amer. Math. Soc., 69 (1963), 766-770. | MR 27 #5086 | Zbl 0143.34701

[7] D. Müller, On the spectral synthesis problem for hypersurfaces of Rn, J. Functional Analysis, 47 (1982), 247-280. | Zbl 0507.43003