×

zbMATH — the first resource for mathematics

Hilbert integrals, singular integrals, and Radon transforms. I. (English) Zbl 0622.42011
This paper is a full exposition of a preliminary announcement by the authors [Proc. Natl. Acad. Sci. USA 80, 7697-7701 (1983; Zbl 0567.42010)]. Here is the contents:
Introduction: singular Radon transforms R; the model case; \(L^ p\) estimates, \(1\leq p<\infty\), for R on a smooth manifold \(\Omega\) of dimension \(\geq 3\); similar estimates for the maximal function M; appendix on \(L^ 2\) boundedness of pseudo-differential operators with operator- valued symbols of the class \(S_{,}\); references.
In order to obtain a reasonable \(L^ p\) theory, is necessary to impose some curvature condition on the hypersurface \(\Omega_ P\) which supports a singular integral density K(P,\(\cdot)\) with its singularity at P. The estimates for the singular Radon transforms draws on three sources: the idea of twisted convolution for the Heisenberg group, suggestive results of Hörmander on oscillatory integrals and the construction of appropriate analytic families of operators.
Singular Radon transforms will be applied, in the next paper of this series, to the Hilbert integral operators as they arise in boundary-value problems, in particular, in the non-coercive case, such as in the \({\bar \partial}\)-Neumann problem for strongly pseudo-convex domains.
Reviewer: R.Vaillancourt

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
58J40 Pseudodifferential and Fourier integral operators on manifolds
47Gxx Integral, integro-differential, and pseudodifferential operators
44A15 Special integral transforms (Legendre, Hilbert, etc.)
42B25 Maximal functions, Littlewood-Paley theory
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Agmon, S., Douglis, A. &Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I.Comm. Pure Appl. Math., 12 (1959), 623–727; II. ibid. Agmon, S., Douglis, A. & Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II.Comm. Pure Appl. Math., 17 (1964), 35–92. · Zbl 0093.10401 · doi:10.1002/cpa.3160120405
[2] Beals, M., Fefferman, C. &Grossman, R., Strictly pseudo-convex domains.Bull. Amer. Math. Soc., 8 (1983), 125–322. · Zbl 0546.32008 · doi:10.1090/S0273-0979-1983-15087-5
[3] Boutet de Monvel, L., Comportement d’un opérateur pseudo-differentiel sur une variété à bord I.J. Analyse Math., 17 (1966), 241–253; II. ibid.Boutet de Monvel, L., Comportement d’un opérateur pseudo-differentiel sur une variété à bord II.J. Analyse Math., 16 (1966), 255–304. · Zbl 0161.07902 · doi:10.1007/BF02788660
[4] Boutet de Monvel, L. &Sjöstrand, J., Sur la singularité des noyaux de Bergman et de Szegö.Astérisque, 34–35 (1976), 123–164. · Zbl 0344.32010
[5] Carleson, L., On convergence and growth of partial sums of Fourier series.Acta Math., 116 (1966), 135–157. · Zbl 0144.06402 · doi:10.1007/BF02392815
[6] Chiang, C., To appear.
[7] Coifman, R. & Meyer, Y.,Au delà des opérateurs pseudo-différentiels, 57 (1978).
[8] Coifman, R. & Weiss, G.,Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Math., 242 (1971). Springer-Verlag. · Zbl 0224.43006
[9] Christ, M., Hilbert transforms along curves I. Nilpotent groups.Ann. of Math., 122 (1985), 575–596. · Zbl 0593.43011 · doi:10.2307/1971330
[10] Eskin, G.,Boundary value problems for elliptic pseudo-differential equations, American Mathematical Society, Providence, R.I., 1981. · Zbl 0458.35002
[11] Fefferman, C., The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math., 26 (1974), 1–66. · Zbl 0289.32012 · doi:10.1007/BF01406845
[12] Fefferman, C. &Stein, E. M.,H p spaces in several variables.Acta Math., 129 (1972), 137–193. · Zbl 0257.46078 · doi:10.1007/BF02392215
[13] Folland, G. & Kohn, J. J.,The Neumann problem for the Cauchy-Riemann complex. Ann. Math. Studies 75 (1972). Princeton University Press. · Zbl 0247.35093
[14] Folland, G. &Stein, E. M., Estimates for the \(\bar \partial _b\) -complex and analysis on the Heisenberg group.Comm. Pure Appl. Math., 27 (1974), 429–522. · Zbl 0293.35012 · doi:10.1002/cpa.3160270403
[15] Folland, G. & Stein, E. M.,Hardy spaces on homogeneous groups. Math. Notes 28 (1982). Princeton University Press. · Zbl 0508.42025
[16] Geller, D. &Stein, E. M., Singular convolution operators on the Heisenberg group.Bull. Amer. Math. Soc., 6 (1982), 99–103. · Zbl 0483.43005 · doi:10.1090/S0273-0979-1982-14977-1
[17] –, Estimates for singular convolution operators on the Heisenberg group.Math. Ann., 267 (1984), 1–15. · Zbl 0537.43005 · doi:10.1007/BF01458467
[18] Greenleaf, A., Pointwise convergence of singular convolution operators. To appear. · Zbl 0533.43006
[19] Greiner, P. & Stein, E. M., A parametrix for the \(\bar \partial\) -Neumann problem. Math. Notes 19 (1977). Princeton University Press. · Zbl 0354.35002
[20] Grossman, A., Loupias, G. &Stein, E. M., An algebra of pseudo-differential operators and quantum mechanics in phase space.Ann. Institut Fourier, 18 (1969), 343–368.
[21] Guillemin, V. & Sternberg, S.,Geometric asymptotics. Amer. Math. Society, 1977.
[22] Guillemin, V. &Uhlmann, G., Oscillatory integrals and singular symbols.Duke Math. J., 48 (1981), 251–267. · Zbl 0462.58030 · doi:10.1215/S0012-7094-81-04814-6
[23] Harvey, R. & Polking, J. The \(\bar \partial\) Neumann kernel in the ball in C n .Trans. Amer. Math. Soc. · Zbl 0578.32030
[24] Hörmander, L.,Linear partial differential operators. Springer-Verlag, 1963. · Zbl 0108.09301
[25] –, Pseudo-differential operators and non-elliptic boundary problems.Ann. of Math., 83 (1966), 129–209. · Zbl 0132.07402 · doi:10.2307/1970473
[26] –, Fourier integral operators I.Acta Math., 127 (1971), 79–183. · Zbl 0212.46601 · doi:10.1007/BF02392052
[27] –, Oscillatory integrals and multipliers onFL p .Ark. Mat., 11 (1973), 1–11. · Zbl 0254.42010 · doi:10.1007/BF02388505
[28] Howe, R., Quantum mechanics and partial differential equations.J. Funct. Anal., 38 (1980), 188–254. · Zbl 0449.35002 · doi:10.1016/0022-1236(80)90064-6
[29] Lieb, I. &Range, M., On integral representations and a priori estimates.Math. Ann., 265 (1983), 221–251. · Zbl 0514.32015 · doi:10.1007/BF01460799
[30] Mauceri, G., Picardello, M. A. &Ricci, F., Twisted convolution, Hardy spaces, and Hörmander multipliers.Supp. Rend. Circ. Mat. Palermo, 1 (1981), 191–202. · Zbl 0472.43006
[31] Melrose, R. &Uhlmann, G., Lagrangian intersection and the Cauchy problem.Comm. Pure Appl. Math., 32 (1979), 483–519. · Zbl 0406.58017 · doi:10.1002/cpa.3160320403
[32] –, Microlocal structure in involutive conical refraction.Duke Math. J., 46 (1979), 571–582. · Zbl 0422.58026 · doi:10.1215/S0012-7094-79-04630-1
[33] Müller, D., Calderón-Zygmund kernels carried by linear subspaces of homogeneous nilpotent Lie algebras,Invent. Math., 73 (1983), 467–489. · Zbl 0521.43009 · doi:10.1007/BF01388440
[34] –, Singular kernels supported by homogeneous submanifolds.J. Reine Angew. Math., 356 (1985), 90–118. · Zbl 0551.43005 · doi:10.1515/crll.1985.356.90
[35] Nagel, A., Stein, E. M. &Wainger, S., Hilbert transforms and maximal functions related to variable curves.Proc. Symp. Pure Math., 35 (1979), 95–98. · Zbl 0463.42008
[36] Phong, D. H., On integral representations for the Neumann operator.Proc. Nat. Acad. Sci. U.S.A., 76 (1979), 1554–1558. · Zbl 0402.35074 · doi:10.1073/pnas.76.4.1554
[37] Phong, D. H. &Stein, E. M., Estimates for the Bergman and Szego projections on strongly pseudoconvex domains.Duke Math. J., 44 (1977), 695–704. · Zbl 0392.32014 · doi:10.1215/S0012-7094-77-04429-5
[38] –, Some further classes of pseudo-differential and singular integral operators arising in boundary value problems I.Amer. J. Math., 104 (1982), 141–172. · Zbl 0526.35079 · doi:10.2307/2374071
[39] Phong, D. H. & Stein, E. M., Singular integrals with kernels of mixed homogeneities.Conference on harmonic analysis in honor of A. Zygmund. Eds. Beckner, W., Calderón, A., Fefferman, R., Jones, P., Wadsworth. 327–339 (1982).
[40] –, Singular integrals related to the Radon transform and boundary value problems.Proc. Nat. Acad. Sci. U.S.A., 80 (1983), 7697–7701. · Zbl 0567.42010 · doi:10.1073/pnas.80.24.7697
[41] Rempel, S. &Schulze, B. W.,Index theory of boundary value problems, Akademie-Verlag, Berlin, 1983. · Zbl 0548.35116
[42] Ricci, F., Calderón-Zygmund kernels on nilpotent Lie groups, inLecture Notes in Math., 908 (1982), 217–227. Springer-Verlag.
[43] Sampson, G., Oscillating kernels that mapH 1 intoL 1.Ark. Math., 18 (1981), 125–144. · Zbl 0473.42013 · doi:10.1007/BF02384686
[44] Sjölin, P., Convergence almost everywhere of certain singular integrals and multiple Fourier series.Ark. Mat., 9 (1971), 65–90. · Zbl 0212.41703 · doi:10.1007/BF02383638
[45] –, Convolution with oscillating kernels.Indiana Univ. Math. J. 30 (1981), 47–56. · Zbl 0453.47032 · doi:10.1512/iumj.1981.30.30004
[46] Stanton, N. K., The solution of the \(\bar \partial\) -Neumann problem in a strictly pseudoconvex Siegel domain.Invent. Math., 65 (1981), 137–174. · Zbl 0482.32005 · doi:10.1007/BF01389299
[47] Stein, E. M.,Singular integrals and differentiability of functions. Princeton University Press, 1970. · Zbl 0207.13501
[48] –, Singular integrals and estimates for the Cauchy-Riemann equations.Bull. Amer. Math. Soc., 79 (1973), 440–445. · Zbl 0257.35040 · doi:10.1090/S0002-9904-1973-13205-7
[49] Stein, E. M. &Wainger, S., Problems in harmonic analysis related to curvature.Bull. Amer. Math. Soc., 84 (1978), 1239–1295. · Zbl 0393.42010 · doi:10.1090/S0002-9904-1978-14554-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.