On boundary values of holomorphic functions on balls

Globevnik, Josip

doi:10.1090/S0002-9939-1982-0647898-9

On boundary values of holomorphic functions on balls
HTML articles powered by AMS MathViewer

by Josip Globevnik

Proc. Amer. Math. Soc. 85 (1982), 61-64

DOI: https://doi.org/10.1090/S0002-9939-1982-0647898-9

PDF | Request permission

Abstract:

It is a result of Agranovski and Valski for which Nagel and Rudin, and Stout have given alternate proofs, that if $B$ is the open unit ball in ${{\mathbf {C}}^n}$ and if $f \in C(\partial B)$ has the property that for every complex line $\Lambda \subset {{\mathbf {C}}^n}$, $f\left | {(\Lambda \cap \partial B)} \right .$ has a continuous extension to $\Lambda \cap \bar B$ which is holomorphic in $\Lambda \cap B$, then $f$ has a continuous extension to $\bar B$ which is holomorphic in $B$. In the paper we give an easier, more geometric proof of this result and then prove the local version of this result.

References

M. L. Agranovskiĭ and R. È. Val′skiĭ, Maximality of invariant algebras of functions, Sibirsk. Mat. Ž. 12 (1971), 3–12 (Russian). MR 0285911
Alexander Nagel and Walter Rudin, Moebius-invariant function spaces on balls and spheres, Duke Math. J. 43 (1976), no. 4, 841–865. MR 425178
Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841

Function theory in the unit ball of

Edgar Lee Stout, The boundary values of holomorphic functions of several complex variables, Duke Math. J. 44 (1977), no. 1, 105–108. MR 437800
Edgar Lee Stout, Analytic continuation and boundary continuity of functions of several complex variables, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), no. 1-2, 63–74. MR 628129, DOI 10.1017/S0308210500032364
B. V. Shabat, Vvedenie v kompleksnyĭ analiz, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0584932
Barnet M. Weinstock, An approximation theorem for $\overline \partial$-closed forms of type $(n,\,n-1)$, Proc. Amer. Math. Soc. 26 (1970), 625–628. MR 265638, DOI 10.1090/S0002-9939-1970-0265638-4