Singular integrals with rough kernels along real-analytic submanifolds in 𝐑³

Fan, Dashan; Guo, Kanghui; Pan, Yibiao

doi:10.1090/S0002-9947-02-03175-6

Singular integrals with rough kernels along real-analytic submanifolds in ${\mathbf {R}}^3$
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by Dashan Fan, Kanghui Guo and Yibiao Pan PDF

Trans. Amer. Math. Soc. 355 (2003), 1145-1165 Request permission

Abstract:

$L^p$ mapping properties will be established in this paper for singular Radon transforms with rough kernels defined by translates of a real-analytic submanifold in $\mathbf {R}^3$.

References

Michael Christ, Hilbert transforms along curves. I. Nilpotent groups, Ann. of Math. (2) 122 (1985), no. 3, 575–596. MR 819558, DOI 10.2307/1971330
Michael Christ, Alexander Nagel, Elias M. Stein, and Stephen Wainger, Singular and maximal Radon transforms: analysis and geometry, Ann. of Math. (2) 150 (1999), no. 2, 489–577. MR 1726701, DOI 10.2307/121088
Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
Colzani, L., Hardy spaces on spheres, Ph.D. Thesis, Washington University, St. Louis, 1982.
William C. Connett, Singular integrals near $L^{1}$, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 163–165. MR 545253
Javier Duoandikoetxea and José L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541–561. MR 837527, DOI 10.1007/BF01388746
Dashan Fan, Restriction theorems related to atoms, Illinois J. Math. 40 (1996), no. 1, 13–20. MR 1386309
R. Fefferman, A note on singular integrals, Proc. Amer. Math. Soc. 74 (1979), no. 2, 266–270. MR 524298, DOI 10.1090/S0002-9939-1979-0524298-3
Dashan Fan and Yibiao Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math. 119 (1997), no. 4, 799–839. MR 1465070
Dashan Fan, Kanghui Guo, and Yibiao Pan, Singular integrals with rough kernels along real-analytic submanifolds in $\mathbf R^n$, Integral Equations Operator Theory 33 (1999), no. 1, 8–19. MR 1664347, DOI 10.1007/BF01203079
Robert M. Hardt, Slicing and intersection theory for chains associated with real analytic varieties, Acta Math. 129 (1972), 75–136. MR 315561, DOI 10.1007/BF02392214
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
Detlef Müller, Singular kernels supported by homogeneous submanifolds, J. Reine Angew. Math. 356 (1985), 90–118. MR 779377, DOI 10.1515/crll.1985.356.90
Javad Namazi, A singular integral, Proc. Amer. Math. Soc. 96 (1986), no. 3, 421–424. MR 822432, DOI 10.1090/S0002-9939-1986-0822432-2
Yibiao Pan, Boundedness of oscillatory singular integrals on Hardy spaces. II, Indiana Univ. Math. J. 41 (1992), no. 1, 279–293. MR 1160914, DOI 10.1512/iumj.1992.41.41016
D. H. Phong and E. M. Stein, Singular integrals related to the Radon transform and boundary value problems, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 24, , Phys. Sci., 7697–7701. MR 728667, DOI 10.1073/pnas.80.24.7697
D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I, Acta Math. 157 (1986), no. 1-2, 99–157. MR 857680, DOI 10.1007/BF02392592
Fulvio Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals, J. Funct. Anal. 73 (1987), no. 1, 179–194. MR 890662, DOI 10.1016/0022-1236(87)90064-4
Fulvio Ricci and Elias M. Stein, Harmonic analysis on nilpotent groups and singular integrals. II. Singular kernels supported on submanifolds, J. Funct. Anal. 78 (1988), no. 1, 56–84. MR 937632, DOI 10.1016/0022-1236(88)90132-2
Fulvio Ricci and Guido Weiss, A characterization of $H^{1}(\Sigma _{n-1})$, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 289–294. MR 545268
E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 196–221. MR 934224
E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 307–355. MR 864375
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295. MR 508453, DOI 10.1090/S0002-9904-1978-14554-6