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Growth Spaces and Growth Norm Estimates for on Convex Domains of Finite Type

Published online by Cambridge University Press:  20 November 2018

Hong Rae Cho
Hong Rae Cho*
Affiliation:
Department of Mathematics, Pusan National University, Pusan 609-735, South Korea e-mail: chohr@pusan.ac.kr
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Abstract

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We consider the growth norm of a measurable function $f$ defined by

$${{\left\| f \right\|}_{-\sigma }}=\text{ess}\,\,\text{sup}\left\{ {{\delta }_{D}}{{\left( z \right)}^{\sigma }}\left| f\left( z \right) \right|:z\in D \right\},$$

where ${{\delta }_{D}}\left( z \right)$ denote the distance from $z$ to $\partial D$. We prove some optimal growth norm estimates for $\bar{\partial }$ on convex domains of finite type.

Keywords

32W0532A2632A36
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 4 , 01 December 2006 , pp. 508 - 525
Copyright
Copyright © Canadian Mathematical Society 2006

References

[AhCh] Ahn, H. and Cho, H. R., Optimal Sobolev estimates for on convex domains of finite type. Math. Z. 244(2003), no. 4, 837–857.Google Scholar
[Al] Alexandre, W., Estimées Ck pour les domaines convexes de type fini de ℂ n . C. R. Acad. Sci. Paris 335(2002), no 1, 23–26.Google Scholar
[Be] Beatrous, F., Estimates for derivatives of holomorphic functions in pseudoconvex domains. Math. Z. 191(1986), no. 1, 91–116.Google Scholar
[BeBu] Beatrous, F. and Burbea, J., Holomorphic Sobolev spaces on the ball. Dissertationes Math. 276(1989).Google Scholar
[BeAn] Berndtsson, B. and Andersson, M., Henkin-Ramirez formulas with weight factors. Ann. Inst. Fourier (Grenoble) 32(1982), 91–110.Google Scholar
[Ch] Cho, H. R., Estimates on the mean growth of Hp functions in convex domains of finite type. Proc. Amer. Math. Soc. 131(2003), no. 8, 2393–2398.Google Scholar
[ChKw] Cho, H. R. and Kwon, E. G., Embedding of Hardy spaces into weighted Bergman spaces in bounded domains with C 2 boundary. Illinois J. Math. 48(2004), no. 3, 747–757.Google Scholar
[Cu] Cumenge, A., Sharp estimates for on convex domains of finite type. Ark. Mat. 39(2001), no. 1, 1–25.Google Scholar
[DFF] Diederich, K., Fischer, B., and Fornæss, J. E., Hölder estimates on convex domains of finite type. Math. Z. 232(1999), no. 1, 43–61.Google Scholar
[DiFi] Diederich, K. and Fischer, B., Holder estimates on lineally convex domains of finite type. Preprint 2003.Google Scholar
[DiFo1] Diederich, K. and Fornæss, J. E., Support functions for convex domains of finite type. Math. Z. 230(1999), no. 1, 145–164.Google Scholar
[DiFo2] Diederich, K. and Fornæss, J. E., Lineally convex domains of finite type: holomorphic support functions. Manuscripta Math. 112(2003), no. 4, 403–431.Google Scholar
[DiMa] Diederich, K. and Mazzilli, E., Zero varieties for the Nevanlinna class on all convex domains of finite type. Nagoya Math. J. 163(2001), 215–227.Google Scholar
[Du] Duren, P. L., Theory of Hp Spaces. Pure and Applied Mathematics 38, Academic Press, New York, 1970.Google Scholar
[Fi] Fischer, B., Lp estimates on convex domains of finite type. Math. Z. 236(2001), no. 2, 401–418.Google Scholar
[He1] Hefer, T., Hölder and Lp estimates for on convex domains of finite type depending on Catlin's multitype. Math. Z. 242(2002), no. 2, 367–398.Google Scholar
[He2] Hefer, T., Extremal bases and Hölder estimates for on convex domains of finite type. Mich. Math. J. 52(2004), no. 3, 573–602.Google Scholar
[Mc] McNeal, J. D., Estimates on the Bergman kernels of convex domains. Adv. Math. 109(1994), no. 1, 108–139.Google Scholar
[McSt] McNeal, J. D. and Stein, E. M., Mapping properties of the Bergman projection on convex domains of finite type. Duke Math. J. 73(1994), no. 1, 177–199.Google Scholar
[Ra1] Range, R. M., On Hölder estimates for u = f on weakly pseudoconvex domains. In: Several Complex Variables. Scuola Norm. Sup. Pisa, Pisa, 1979, pp. 247–267.Google Scholar
[Ra2] Range, R. M., Holomorphic Functions and Integral Representations in Several Complex Variables. Graduate Texts in Mathematics 108, Springer-Verlag, Berlin, 1986.Google Scholar
[St] Stein, E. M., Boundary Behavior of Holomorphic Functions of Several Complex Variables. Mathematical Notes 11, Princeton Univ. Press, Princeton, NJ, 1972.Google Scholar
[Wa] Wang, W., Hölder regularity for on the convex domains of finite strict type. Pacific J. Math. 198(2001), no. 1, 235–256.Google Scholar