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

Straube, Emil J.

doi:10.1090/S0002-9947-1989-0943308-3

Interpolation between Sobolev and between Lipschitz spaces of analytic functions on starshaped domains
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by Emil J. Straube

Trans. Amer. Math. Soc. 316 (1989), 653-671

DOI: https://doi.org/10.1090/S0002-9947-1989-0943308-3

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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

David E. Barrett, Regularity of the Bergman projection and local geometry of domains, Duke Math. J. 53 (1986), no. 2, 333–343. MR 850539, DOI 10.1215/S0012-7094-86-05321-4
R. P. Boas Jr., Inequalities for monotonic series, J. Math. Anal. Appl. 1 (1960), 121–126. MR 117465, DOI 10.1016/0022-247X(60)90032-9
Frank Beatrous Jr., Estimates for derivatives of holomorphic functions in pseudoconvex domains, Math. Z. 191 (1986), no. 1, 91–116. MR 812605, DOI 10.1007/BF01163612

Interpolation spaces

Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 0230022
Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
Jacqueline Détraz, Classes de Bergman de fonctions harmoniques, Bull. Soc. Math. France 109 (1981), no. 2, 259–268 (French, with English summary). MR 623793
Björn E. J. Dahlberg, On the Poisson integral for Lipschitz and $C^{1}$-domains, Studia Math. 66 (1979), no. 1, 13–24. MR 562447, DOI 10.4064/sm-66-1-13-24
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Peter Henrici, Applied and computational complex analysis. Vol. 2, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions. MR 0453984
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Steven G. Krantz, Intrinsic Lipschitz classes on manifolds with applications to complex function theory and estimates for the $\bar \partial$ and $\bar \partial _{b}$ equations, Manuscripta Math. 24 (1978), no. 4, 351–378. MR 496755, DOI 10.1007/BF01168882
Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
Ewa Ligocka, Estimates in Sobolev norms $\|\cdot \|^s_p$ for harmonic and holomorphic functions and interpolation between Sobolev and Hölder spaces of harmonic functions, Studia Math. 86 (1987), no. 3, 255–271. MR 917051, DOI 10.4064/sm-86-3-255-271
Ewa Ligocka, On the reproducing kernel for harmonic functions and the space of Bloch harmonic functions on the unit ball in $\textbf {R}^n$, Studia Math. 87 (1987), no. 1, 23–32. MR 924758, DOI 10.4064/sm-87-1-23-32
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580