Boundedness of certain oscillatory singular integrals

Dashan Fan; Yibiao Pan

ArticleOriginal scientific text

Title

Boundedness of certain oscillatory singular integrals

Authors ¹, ²

Affiliations

epartment of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WIsconsin 53201, U.S.A.
Department of Mathematical and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, U.S.A.

Abstract

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^{i Φ (x)} K (x)

, K(x) is a Calderón-Zygmund kernel, and Φ satisfies certain growth conditions.

S. Chanillo, Weighted norm inequalities for strongly singular convolution operators, Trans. Amer. Math. Soc. 281 (1984), 77-107.
S. Chanillo and M. Christ, Weak (1,1) bounds for oscillatory singular integrals, Duke Math. J. 55 (1987), 141-155.
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.
R. Coifman, A real variable characterization of $H^{p}$ , Studia Math. 51 (1974), 269-274.
R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
D. Fan, An oscillating integral in the Besov space $B^{1}, 0_{0} (ℝ^{n})$ , submitted for publication.
C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36.
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, ibid. 129 (1972), 137-193.
Y. Hu and Y. Pan, Boundedness of oscillatory singular integrals on Hardy spaces, Ark. Mat. 30 (1992), 311-320.
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.
Y. Pan, Uniform estimates for oscillatory integral operators, J. Funct. Anal. 100 (1991), 207-220.
Y. Pan, Hardy spaces and oscillatory singular integrals, Rev. Mat. Iberoamericana 7 (1991), 55-64.
Y. Pan, Boundedness of oscillatory singular integrals on Hardy spaces: II, Indiana Univ. Math. J. 41 (1992), 279-293.
D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals and Radon transforms I, Acta Math. 157 (1986), 99-157.
F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I, J. Funct. Anal. 73 (1987), 179-194.
G. Sampson, Oscillating kernels that map $H^{1}$ into $L^{1}$ , Ark. Mat. 18 (1981), 125-144.
P. Sjölin, Convolution with oscillating kernels on $H^{p}$ spaces, J. London Math. Soc. 23 (1981), 442-454.
P. Sjölin, Convolution with oscillating kernels, Indiana Univ. Math. J. 30 (1981), 47-55.
M. Spivak, Calculus on Manifolds, Addison-Wesley, New York, N.Y., 1992.
E. M. Stein, Oscillatory integrals in Fourier analysis, in: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., 1986, pp. 307-355.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, N.J., 1993.
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.

Title

Boundedness of certain oscillatory singular integrals

Affiliations

Abstract

Bibliography