Browse > Article
http://dx.doi.org/10.4134/BKMS.b180684

ON HÖLDER ESTIMATES FOR CAUCHY TRANSFORMS ON CONVEX DOMAINS IN ℂ2  

Ha, Ly Kim (Faculty of Mathematics and Computer Science University of Science Vietnam National University)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.4, 2019 , pp. 929-937 More about this Journal
Abstract
The main purpose of this paper is to establish $H{\ddot{o}}lder$ estimates for the Cauchy transform in a class of finite/infinite type convex domains in $\mathbb{C}^2$.
Keywords
Cauchy transform; Lipschitz spaces; convex domain; infinite type; Cauchy-$Fantappi{\grave{e}}$ form;
Citations & Related Records
연도 인용수 순위
  • Reference
1 K. Adachi and H. R. Cho, Some Lipschitz regularity of the Cauchy transform on a convex domain in $C^2$ with real analytic boundary, Commun. Korean Math. Soc. 11 (1996), no. 4, 975-981.
2 P. Ahern and R. Schneider, Holomorphic Lipschitz functions in pseudoconvex domains, Amer. J. Math. 101 (1979), no. 3, 543-565. https://doi.org/10.2307/2373797   DOI
3 S.-C. Chen, Real analytic boundary regularity of the Cauchy kernel on convex domains, Proc. Amer. Math. Soc. 108 (1990), no. 2, 423-432. https://doi.org/10.2307/2048291   DOI
4 L. K. Ha, Tangential Cauchy-Riemann equations on pseudoconvex boundaries of finite and infinite type in $\mathbb{C}^2$, Results Math. 72 (2017), no. 1-2, 105-124. https://doi.org/10.1007/s00025-016-0630-z   DOI
5 L. Lanzani and E. M. Stein, Cauchy-type integrals in several complex variables, Bull. Math. Sci. 3 (2013), no. 2, 241-285. https://doi.org/10.1007/s13373-013-0038-y   DOI
6 L. Lanzani and E. M. Stein, The Cauchy integral in $\mathbb{C}^n$ for domains with minimal smoothness, Adv. Math. 264 (2014), 776-830. https://doi.org/10.1016/j.aim.2014.07.016   DOI
7 L. Lanzani and E. M. Stein, The Cauchy-Szeg}o projection for domains in $\mathbb{C}^n$ with minimal smoothness, Duke Math. J. 166 (2017), no. 1, 125-176. https://doi.org/10.1215/00127094-3714757   DOI
8 L. Lanzani and E. M. Stein, The role of an integration identity in the analysis of the Cauchy-Leray transform, Sci. China Math. 60 (2017), no. 11, 1923-1936. https://doi.org/10.1007/s11425-017-9115-5   DOI
9 J. Leray, Le calcul differentiel et integral sur une variete analytique complexe. (Probleme de Cauchy. III), Bull. Soc. Math. France 87 (1959), 81-180.   DOI
10 E. Ligocka, The Holder continuity of the Bergman projection and proper holomorphic mappings, Studia Math. 80 (1984), no. 2, 89-107. https://doi.org/10.4064/sm-80-2-89-107   DOI
11 E. Ligocka, The regularity of the weighted Bergman projections, in Seminar on deformations (Lodz/Warsaw, 1982/84), 197-203, Lecture Notes in Math., 1165, Springer, Berlin, 1985. https://doi.org/10.1007/BFb0076154   DOI
12 E. Ligocka, On the Forelli-Rudin construction and weighted Bergman projections, Studia Math. 94 (1989), no. 3, 257-272. https://doi.org/10.4064/sm-94-3-257-272   DOI
13 T. V. Khanh, Supnorm and f-Holder estimates for $\bar{\partial}$ on convex domains of general type in $\mathbb{C}^2$, J. Math. Anal. Appl. 403 (2013), no. 2, 522-531. https://doi.org/10.1016/j.jmaa.2013.02.045   DOI
14 W. Koppelman, The Cauchy integral for functions of several complex variables, Bull. Amer. Math. Soc. 73 (1967), 373-377. https://doi.org/10.1090/S0002-9904-1967-11757-9   DOI
15 W. Koppelman, The Cauchy integral for differential forms, Bull. Amer. Math. Soc. 73 (1967), 554-556. https://doi.org/10.1090/S0002-9904-1967-11744-0   DOI
16 J. Verdera, $L^{\infty}$-continuity of Henkin operators solving $\bar{\partial}$ in certain weakly pseudoconvex domains of $\mathbb{C}^2$, Proc. Roy. Soc. Edinburgh Sect. A 99 (1984), no. 1-2, 25-33. https://doi.org/10.1017/S0308210500025932   DOI
17 R. M. Range, Holder estimates for $\bar{\partial}$ on convex domains in $C^2$ with real analytic boundary, in Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975), 31-33, Amer. Math. Soc., Providence, RI, 1977.
18 R. M. Range, The Caratheodory metric and holomorphic maps on a class of weakly pseudoconvex domains, Pacific J. Math. 78 (1978), no. 1, 173-189. http://projecteuclid.org/euclid.pjm/1102806309   DOI
19 R. M. Range, On Holder estimates for $\bar{\partial}u$ = f on weakly pseudoconvex domains, in Several complex variables (Cortona, 1976/1977), 247-267, Scuola Norm. Sup. Pisa, Pisa, 1978.
20 R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Graduate Texts in Mathematics, 108, Springer-Verlag, New York, 1986. https://doi.org/10.1007/978-1-4757-1918-5
21 N. Kerzman and E. M. Stein, The Szego kernel in terms of Cauchy-Fantappie kernels, Duke Math. J. 45 (1978), no. 2, 197-224. http://projecteuclid.org/euclid.dmj/1077312816   DOI