Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Interpolation between Sobolev and between Lipschitz spaces of analytic functions on starshaped domains

Author: Emil J. Straube
Journal: Trans. Amer. Math. Soc. 316 (1989), 653-671
MSC: Primary 46E15; Secondary 32A07, 46E35, 46M35
MathSciNet review: 943308
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that on a starshaped domain $ \Omega $ in $ {\operatorname{C} ^n}$ (actually on a somewhat larger, biholomorphically invariant class) the $ {\mathcal{L}^p}$-Sobolev spaces of analytic functions form an interpolation scale for both the real and complex methods, for each $ p,\;0 < p \leqslant \infty $. The case $ p = \infty $ gives the Lipschitz scale; here the functor $ {(,)^{[\theta ]}}$ has to be considered (rather than $ {(,)_{[\theta ]}}$).

References [Enhancements On Off] (What's this?)

  • [Ba] D. Barrett, Regularity of the Bergman projection and local geometry of domains, Duke Math. J. 53 (1986), 333-343. MR 850539 (87j:32073)
  • [B] R. P. Boas, Jr., Inequalities for monotonic series, J. Math. Anal. Appl. 1 (1960), 121-126. MR 0117465 (22:8244)
  • [Be] F. Beatrous, Estimates for derivatives of holomorphic functions in pseudoconvex domains, Math. Z. 191 (1986), 91-116. MR 812605 (87b:32033)
  • [B-L] J. Bergh and J. Löfström, Interpolation spaces, Grundlehren Math. Wiss., vol. 223, Springer, Berlin, 1976.
  • [Bu-Be] P. Butzer and H. Berens, Semi-groups of operators and approximation, Grundlehren Math. Wiss., vol. 145, Springer, Berlin, 1967. MR 0230022 (37:5588)
  • [C-W] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 0447954 (56:6264)
  • [D] J. Detraz, Classes de Bergman de fonctions harmoniques, Bull. Soc. Math. France 109 (1981), 259-268. MR 623793 (83m:31006)
  • [Da] B. Dahlberg, On the Poisson integral for Lipschitz and $ {C^1}$-domains, Studia Math. 66 (1979), 13-24. MR 562447 (81g:31007)
  • [Du] P. Duren, Theory of $ {H^p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [F-S] Ch. Fefferman and E. Stein, $ {H^p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • [G] J. Garnett, Bounded analytic functions, Academic Press, New York, 1981. MR 628971 (83g:30037)
  • [H] P. Henrici, Applied and computational complex analysis, vol. 2, Wiley, New York, 1977. MR 0453984 (56:12235)
  • [H-Ph] E. Hille and R. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 0089373 (19:664d)
  • [K1] S. Krantz, Intrinsic Lipschitz classes on manifolds with applications to complex function theory and estimates for the $ \overline \partial $ and $ {\overline \partial _b}$ equations, Manuscripta Math. 24 (1978), 351-378. MR 496755 (80b:26020)
  • [K2] -, Function theory of several complex variables, Wiley, New York, 1982. MR 635928 (84c:32001)
  • [K-P-S] S. Krein, J. Petunin, and E. Semenov, Interpolation of linear operators, Transl. Math. Monographs, vol. 54, Amer. Math. Soc., Providence, R. I., 1982. MR 649411 (84j:46103)
  • [Li1] E. Ligocka, Estimates in Sobolev norms $ \vert\vert\;\;\vert\vert _p^s$ for harmonic and holomorphic functions and interpolation between Sobolev and Hölder spaces of harmonic functions, Studia Math. 86 (1987), 225-271. MR 917051 (88k:46034)
  • [Li2] -, On the space of Bloch harmonic functions and interpolation of spaces of harmonic and holomorphic functions, Studia Math. 87 (1987), 223-238. MR 927506 (89f:46054b)
  • [M] V. Maz'ja, Sobolev spaces, Springer, Berlin, 1985. MR 817985 (87g:46056)
  • [Ste] E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. MR 0290095 (44:7280)
  • [T] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, 1978. MR 503903 (80i:46032b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46E15, 32A07, 46E35, 46M35

Retrieve articles in all journals with MSC: 46E15, 32A07, 46E35, 46M35

Additional Information

Keywords: Complex interpolation, real interpolation, Sobolev spaces of analytic functions, Lipschitz spaces of analytic functions, starshaped domains
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society