Boundedness of certain oscillatory singular integrals

Fan, Dashan ; Pan, Yibiao

Studia Mathematica, Tome 113 (1995), p. 105-116 / Harvested from The Polish Digital Mathematics Library

Résumé

We prove the $L^p$ and $H^1$ boundedness of oscillatory singular integral operators defined by Tf = p.v.Ω∗f, where $Ω(x) = e^{iΦ(x)}K(x)$, K(x) is a Calderón-Zygmund kernel, and Φ satisfies certain growth conditions.

Détail

Publié le : 1995-01-01

Zbl 0886.42008

EUDML-ID : urn:eudml:doc:216182

Bibliographie

[00000] [1] S. Chanillo, Weighted norm inequalities for strongly singular convolution operators, Trans. Amer. Math. Soc. 281 (1984), 77-107. | Zbl 0531.42005

[00001] [2] S. Chanillo and M. Christ, Weak (1,1) bounds for oscillatory singular integrals, Duke Math. J. 55 (1987), 141-155. | Zbl 0667.42007

[00002] [3] S. Chanillo, D. Kurtz and G. Sampson, Weighted weak (1,1) and weighted $L^p$ estimates for oscillating kernels, Trans. Amer. Math. Soc. 295 (1986), 127-145. | Zbl 0594.42007

[00003] [4] R. Coifman, A real variable characterization of $H^p$, Studia Math. 51 (1974), 269-274. | Zbl 0289.46037

[00004] [5] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. | Zbl 0358.30023

[00005] [6] D. Fan, An oscillating integral in the Besov space $B^1, 0_0 (ℝ^n)$, submitted for publication.

[00006] [7] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. | Zbl 0188.42601

[00007] [8] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, ibid. 129 (1972), 137-193. | Zbl 0257.46078

[00008] [9] Y. Hu and Y. Pan, Boundedness of oscillatory singular integrals on Hardy spaces, Ark. Mat. 30 (1992), 311-320. | Zbl 0779.42007

[00009] [10] W. B. Jurkat and G. Sampson, The complete solution to the $(L^p, L^q)$ mapping problem for a class of oscillating kernels, Indiana Univ. Math. J. 30 (1981), 403-413. | Zbl 0507.47013

[00010] [11] Y. Pan, Uniform estimates for oscillatory integral operators, J. Funct. Anal. 100 (1991), 207-220. | Zbl 0735.45010

[00011] [12] Y. Pan, Hardy spaces and oscillatory singular integrals, Rev. Mat. Iberoamericana 7 (1991), 55-64. | Zbl 0728.42013

[00012] [13] Y. Pan, Boundedness of oscillatory singular integrals on Hardy spaces: II, Indiana Univ. Math. J. 41 (1992), 279-293. | Zbl 0779.42008

[00013] [14] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals and Radon transforms I, Acta Math. 157 (1986), 99-157. | Zbl 0622.42011

[00014] [15] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I, J. Funct. Anal. 73 (1987), 179-194. | Zbl 0622.42010

[00015] [16] G. Sampson, Oscillating kernels that map $H^1$ into $L^1$, Ark. Mat. 18 (1981), 125-144. | Zbl 0473.42013

[00016] [17] P. Sjölin, Convolution with oscillating kernels on $H^p$ spaces, J. London Math. Soc. 23 (1981), 442-454. | Zbl 0426.46034

[00017] [18] P. Sjölin, Convolution with oscillating kernels, Indiana Univ. Math. J. 30 (1981), 47-55. | Zbl 0419.47020

[00018] [19] M. Spivak, Calculus on Manifolds, Addison-Wesley, New York, N.Y., 1992.

[00019] [20] E. M. Stein, Oscillatory integrals in Fourier analysis, in: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., 1986, pp. 307-355.

[00020] [21] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, N.J., 1993. | Zbl 0821.42001

[00021] [22] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971. | Zbl 0232.42007