Constructing continuous functions holomorphic to a curve
The author presents new interesting proofs of a theorem of \textit{A. Browder} and \textit{J. Wermer} [J. Math. Mech. 12, 119-130 (1963; Zbl 0113.314)], a theorem of A. Davie and a theorem of \textit{T. Gamelin} and \textit{J. Garnett} [J. Funct. Anal. 8, 360-404 (1971; Zbl 0223.30056)] on the characterization of those compact sets \(K\subset \hat C\) for which the algebras \(A_ K\) of all bounded analytic functions on \(\hat C\setminus K\) which have a continuous extension to the closure of \(\hat C\setminus K\) are Dirichlet algebras. The author's methods consist in replacing the use of functional analysis in the original proofs by the explicit construction of bounded solutions of certain \({\bar \partial}\)-problems. They are of a geometrical nature and are intimately related to recent developments in the theory of \(H^{\infty}\) and BMO; see e.g., \textit{J. B. Garnett} and \textit{P. W. Jones} [Acta Math. 155, 27-40 (1985; Zbl 0578.30043)].
bounded solutions of certain \({\bar \partial }\)- problems
scientific article; zbMATH DE number 4109382
scientific article