Harmonic Analysis Techniques in Several Complex Variables

Authors

  • Loredana Lanzani Department of Mathematics, Syracuse University

DOI:

https://doi.org/10.6092/issn.2240-2829/4767

Keywords:

Cauchy Integral, T(1)-theorem, space of homogeneous type, Leray-Levi measure, Szegő projection, Bergman projection, Hardy space, Bergman space, Lebesgue space, pseudoconvex domain, minimal smoothness

Abstract

We give a survey of recent joint work with E.M. Stein (Princeton University) concerning the application of suitable versions of the T(1)-theorem technique to the study of orthogonal projections onto the Hardy and Bergman spaces of holomorphic functions for domains with minimal boundary regularity.

References

Ahern P. and Schneider R., A smoothing property of the Henkin and Szegő projections, Duke Math. J. 47 (1980), 135-143.

Ahern P. and Schneider R., The boundary behavior of Henkin's kernel, Pacific J. Math. 66 (1976), 9-14.

Andersson M., Passare M. and Sigurdsson R., Complex convexity and analytic functionals, Birkhäuser, Basel (2004).

Barrett D. E., Irregularity of the Bergman projection on a smooth bounded domain, Ann. of Math. 119 (1984) 431-436.

Barrett D. E., Behavior of the Bergman projection on the Diederich-Fornæss worm, Acta Math. 168 (1992) 1-10.

Barrett D. and Lanzani L., The Leray transform on weighted boundary spaces for convex Reinhardt domains, J. Funct. Analysis, 257 (2009), 2780-2819.

Barrett D. and Sahtoglu S., Irregularity of the Bergman projection on worm domains in Cn, Michigan Math. J. 61 no. 2 (2012), 187?198.

Barrett D. and Vassiliadou S., The Bergman kernel on the intersection of two balls in C2, Duke Math. J. 120 no. 2 (2003), 441-467.

Bekollé D., Projections sur des espaces de fonctions holomorphes dans des domains planes, Can. J. Math. XXXIII (1986), 127-157.

Bekollé D. and Bonami A., Inegalites a poids pour le noyau de Bergman, C. R. Acad. Sci. Paris Ser. A-B 286 no. 18 (1978), A775-A778.

Bell S. and Ligocka E., A simplification and extension of Fefferman's theorem on biholomorphic mappings, Invent. Math. 57 (1980) no. 3, 283-289.

Berndtsson B., Cauchy-Leray forms and vector bundles. Ann. Sci. Èc. Norm. Super. 24 no. 4 (1991) 319-333.

Berndtsson B., Weighted integral formulas in several complex variables Stockholm, 1987/1988, in: Math. Notes vol. 38, Princeton univ. Press, Princeton NJ 1993, pp. 160-187.

Bolt M., The Möbius geometry of hypersurfaces, Michigan Math. J. 56 (2008), no. 3, 603-622.

Bolt M., A geometric characterization: complex ellipsoids and the Bochner-Martinelli kernel, Illinois J. Math., 49 (2005), 167 - 184.

Bonami A. and Lohoué N., Projecteurs de Bergman et Szegő pour une classe de domaines faiblement pseudo-convexes et estimations Lp Compositio Math. 46, no. 2 (1982), 159-226.

Calderón A. P, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. 74 no. 4, (1977) 1324-1327.

Calderón A. P. and Zygmund A. Local properties of solutions of partial differential equations, Studia Math. 20 (1961), 171-225.

Charpentier, P. and Dupain, Y. Estimates for the Bergman and Szegő Projections for pseudo-convex domains of finite type with locally diagonalizable Levi forms, Publ. Mat. 50 (2006), 413-446.

Chen S.-C. and Shaw M.-C. Partial differential equations in several complex variables, Amer. Math. Soc., Providence, (2001).

Christ, M. A T(b)-theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990) no. 2, 601-628.

Christ, M. Lectures on singular integral operators CBMS Regional Conf. Series 77, American Math. Soc., Providence, (1990).

Coifman R., McIntosh A. and Meyer Y., L'intègrale de Cauchy dèfinit un opèrateur bornè sur L2 pour les courbes lipschitziennes Ann. of Math. 116 (1982) no. 2, 361-387.

Cumenge A., Comparaison des projecteurs de Bergman et Szegő et applications, Ark. Mat. 28 (1990), 23-47.

David G., Opérateurs intégraux singuliers sur certain courbes du plan complexe, Ann. Scient. éc. Norm. Sup. 17 (1984), 157-189.

David G., Journé J.L., A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. 120 (1984), 371-397.

David G., Journé J.L. and Semmes, S. Oprateurs de Caldern-Zygmund, fonctions para-accrtives et interpolation, Rev. Mat. Iberoamericana 1 (1985) no. 4, 1-56.

Duren P. L., Theory of Hp Spaces, Dover, Mineola (2000).

Ehsani, D. and Lieb I., Lp-estimates for the Bergman projection on strictly pseudo-convex nonsmooth domains Math. Nachr. 281 (2008) 916-929.

Fefferman C., The Bergman kernel and biholomorphic mappings of pseudo-convex domains, Invent. Math. 26 (1974), 1-65.

Federer H., Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc., pp. xiv+676, ISBN 978-3-540-60656-7.

Folland G. B. and Kohn J. J. The Neumann problem for the Cauchy-Riemann complex, Ann. Math. Studies 75, Princeton U. Press, Princeton, 1972.

Francsics G. and Hanges N., Explicit formulas for the Szegő kernel on certain weakly pseudoconvex domains, Proc. Amer. Math. Soc. 123 no. 10 (1995), 3161-3168.

Francsics G. and Hanges N., Trèves curves and the Szegő kernel, Indiana Univ. Math. J. 47 no. 3 (1998), 995-1009.

Francsics G. and Hanges N., The Bergman kernel of complex ovals and multivariable hypergeometric functions, J. Funct. Anal. 142 no. 2 (1996), 494-510.

Francsics G. and Hanges N., Analytic regularity for the Bergman kernel Journées équations aux dérivées partielles (Saint-Jean-des-Monts, 1998), Exp. No. V, 11 pp., Univ. Nantes, Nantes 1998.

Francsics G. and Hanges N., Analytic singularities of the Bergman kernel for tubes, Duke Math. J. 108 no. 3 (2001), 539-580.

Halfpap J., Nagel A. and Weinger S., The Bergman and Szegő kernels near points of infinite type, Pacific J. Math. 246 no. 1 (2010) 75-128.

Hanges N., Explicit formulas for the Szegő kernel for some domains in C2, J. Func. Analysis 88 (1990), 153-165.

Hansson T., On Hardy spaces in complex ellipsoids, Ann. Inst. Fourier (Grenoble) 49 (1999), 1477-1501.

Hedenmalm H., The dual of a Bergman space on simply connected domains, J. d' Analyse 88 (2002), 311-335.

Henkin G., Integral representations of functions holomorphic in strictly pseudo-convex domains and some applications Mat. Sb. 78 (1969) 611-632. Engl. Transl.: Math. USSR Sb. 7 (1969) 597- 616.

Henkin G. M., Integral representations of functions holomorphic in strictly pseudo-convex domains and applications to the @-problem Mat. Sb. 82 (1970) 300- 308. Engl. Transl.: Math. USSR Sb. 11 (1970) 273-281.

Henkin G. M. and Leiterer J., Theory of functions on complex manifolds, Birkhäuser, Basel, (1984).

Hörmander, L. Notions of convexity, Birkhäuser, Basel, (1994).

Hörmander, L. An introduction to complex analysis in several variables, North Holland Math. Library, North Holland Publishing Co., Amsterdam (1990).

Kenig C., Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference series in Mathematics 83, American Math. Soc., Providence (RI) (1994) 0-8218-0309-3.

Kerzman N., Singular integrals in complex analysis, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 2, pp. 3?41, Proc. Sympos. Pure Math., XXXV, Part 2, pp. 3-41, Amer. Math. Soc., Providence, (RI) (1979).

Kerzman, N. and Stein E. M., The Cauchy-Szegő kernel in terms of the Cauchy-Fantappié kernels, Duke Math. J. 25 (1978), 197-224.

Kerzman N. and Stein E. M., The Cauchy kernel, the Szegő kernel, and the Riemann mapping function, Math. Ann. 236 (1978), 85 - 93.

Koenig K. D., Comparing the Bergman and Szegő projections on domains with subelliptic boundary Laplacian Math. Ann. 339 (2007), 667693.

Koenig K. D., An analogue of the Kerzman-Stein formula for the Bergman and Szegő projections, J. Geom. Anal. 19 (2009), 81863.

Koenig, K. and Lanzani, L. Bergman vs. Szegő via Conformal Mapping, Indiana Univ. Math. J. 58, no. 2 (2009), 969-997.

Krantz S., Canonical kernels versus constructible kernels, preprint. ArXiv: 1112.1094.

Krantz S. Function theory of several complex variables, 2nd ed., Amer. Math. Soc., Providence, (2001).

Krantz S. and Peloso M., The Bergman kernel and projection on non-smooth worm domains, Houston J. Math. 34 (2008), 873-950.

Lanzani L., Szegő projection versus potential theory for non-smooth planar domains, Indiana U. Math. J. 48 no. 2 (1999), 537 - 555.

Lanzani L., Cauchy transform and Hardy spaces for rough planar domains, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998), 409?428, Contemp. Math. 251, Amer. Math. Soc., Providence, RI (2000).

Lanzani L. and Stein E. M., Cauchy-Szegő and Bergman projections on non-smooth planar domains, J. Geom. Analysis 14 (2004), 63-86.

Lanzani L. and Stein E. M., The Bergman projection in Lp for domains with minimal smoothness, Illinois J. Math. 56 (1) (2013) 127-154.

Lanzani L. and Stein E. M. Cauchy-type integrals in several complex variables, Bull. Math. Sci. 3 (2) (2013), 241-285, DOI: 10.1007/s13373-013-0038-y.

Lanzani L. and Stein E. M. The Cauchy integral in Cn for domains with minimal smoothness, Advances in Math. 264 (2014) 776-830, DOI: dx.doi.org/10.1016/j.aim.2014.07.016).

Lanzani L. and Stein E. M. The Szegő projection in Lp for domains with minimal smoothness, submitted for publication.

Lanzani L. and Stein E. M. Holomorphic Hardy space representations for domains with minimal smoothness, in preparation.

J. Leray, Le calcul différentiel et intégral sur une variété analytique complexe (Problème de Cauchy, III), Bull. Soc. Math. France 87 (1959) 81-180.

Ligocka E., The Hölder continuity of the Bergman projection and proper holomorphic mappings, Studia Math. 80 (1984), 89-107.

Martineau M. Sur la notion d'ensemble fortement linèellement convexe, An. Acad. Brasil. Ci. 40 (1968), 427-435.

Mattila P., Melnikov M. and Verdera J., The Cauchy integral, analytic capacity, and uniform rectifiability Ann. Math. 144 (2) (1996), 127-136.

McNeal J., Boundary behavior of the Bergman kernel function in C2, Duke Math. J. 58 no. 2 (1989), 499-512.

McNeal, J., Estimates on the Bergman kernel of convex domains, Adv.in Math. 109 (1994) 108-139.

McNeal J. and Stein E. M., Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. J. 73 no. 1 (1994), 177-199.

McNeal J. and Stein E. M., The Szegő projection on convex domains, Math. Zeit. 224 (1997), 519-553.

Melnikov M. and Verdera J., A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs Intern. Math. Res. Notices 7 (1995) 325-331.

Meyer Y., Ondelettes et Opèrateurs II Opèrateurs de Calderon-Zygmund, Actualitès Mathèmatiques, Hermann (Paris), 1990, pp. i - xii and 217- 384. ISBN: 2-7056-6126-7.

Meyer Y. and Coifman R., Ondelettes et Opèrateurs III Opèrateurs multilinèaires Actualitès Mathèmatiques, Hermann (Paris), 1991, pp. i-xii and 383-538. ISBN: 2-7056-6127-1.

Muscalu C. and Schlag W., Classical and multilinear harmonic analysis, II, Cambridge U. Press, Cambrdge, 2013.

Nagel A. and Pramanik M., Diagonal estimates for the Bergman kernel for weakly pseudoconvex domains of monomial type, preprint.

Nagel A., Rosay J.-P., Stein E. M. and Wainger S., Estimates for the Bergman and Szegő kernels in C2, Ann. of Math. 129 no. 2 (1989), 113-149.

Phong D. and Stein E. M., Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains, Duke Math. J. 44 no.3 (1977), 695-704.

Polovinkin E. S., Strongly convex analysis, Mat. Sb. 187 (1996) 103 - 130 (in Russian); translation in Sb. Math. 187 (1996) 259 - 286.

Ramirez E. Ein divisionproblem und randintegraldarstellungen in der komplexen analysis Math. Ann. 184 (1970) 172-187.

Range M., Holomorphic functions and integral representations in several complex variables, Springer Verlag, Berlin, 1986.

Range R. M., An integral kernel for weakly pseudoconvex domains, Math. Ann. 356 (2013), 793-808.

Rotkevich A. S., The Cauchy-Leray-Fantappi integral in linearly convex domains (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 401 (2012), Issledovaniya po Lineinym Operatoram i Teorii Funktsii 40, 172-188, 201; translation in J. Math. Sci. (N. Y.) 194 (2013), no. 6, 693-702.

Rotkevich A. S., The Aizenberg formula in nonconvex domains and some of its applications (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 389 (2011), Issledovaniya po Lineinym Operatoram i Teorii Funktsii. 38, 206-231, 288; translation in J. Math. Sci. (N. Y.) 182 (2012), no. 5, 699-713.

Rudin W. Function Theory in the unit ball of Cn, Springer-Verlag, Berlin (1980).

Stein E. M. Boundary behavior of holomorphic functions of several complex variables, Princeton University Press, Princeton, (1972).

Stein E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy, Princeton Mathematical series 43. Monographs in Harmonic Analysis III. Princeton U. Press, Princeton (NJ) (1993) ISBN: 0-691-03216-5.

Stout E. L., Hp-functions on strictly pseudoconvex domains, American J. Math. 98 no. 3 (1976), 821-852.

Tolsa X., Analytic capacity, rectifiability, and the Cauchy integral, International Congress of Mathematicians, Vol. II, 1505 - 1527, Eur. Math. Soc. Zürich 2006.

Tolsa X., Painlevé's problem and the semiadditivity of analytic capacity, Acta Math. 194 no. 1 (2003), 105-149,

Zeytuncu Y., Lp-regularity of weighted Bergman projections, Trans. AMS 365 (2013), 2959-2976.

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Published

2014-12-30

How to Cite

Lanzani, L. (2014). Harmonic Analysis Techniques in Several Complex Variables. Bruno Pini Mathematical Analysis Seminar, 5(1), 83–110. https://doi.org/10.6092/issn.2240-2829/4767

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