On a theorem of Stein

Krantz, Steven G.

doi:10.1090/S0002-9947-1990-0964899-0

On a theorem of Stein
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by Steven G. Krantz

Trans. Amer. Math. Soc. 320 (1990), 625-642

DOI: https://doi.org/10.1090/S0002-9947-1990-0964899-0

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Abstract:

In this paper the Kobayashi metric on a domain in ${{\mathbf {C}}^n}$ is used to define a new function space. Elements of this space belong to a nonisotropic Lipschitz class. It is proved that if $f$ is holomorphic on the domain and in the classical Lipschitz space ${\Lambda _\alpha }$ then in fact $f$ is in the new function space. The result contains the original result of Stein on this subject and provides the optimal result adapted to any domain. In particular, it recovers the Hartogs extension phenomenon in the category of Lipschitz spaces.

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