On pluriharmonic interpolation. (English) Zbl 0648.32009

Rosay, Jean-Pierre; Stout, Edgar Lee

Math. Scand. 63, No. 2, 268-281 (1988).

Let D be a bounded strongly pseudoconvex domain in \({\mathbb{C}}^ n \)with bD of class \(C^ 2\). Call a compact subset E of bD a pluriharmonic interpolation set if each continuous function on E is the restriction to E of a function continuous on the closure of D and pluriharmonic on D. It is shown that smooth submanifolds of dimension at least two and possibly with boundary of bD that are pluriharmonic interpolation sets are complex-tangential and so are peak-interpolation sets for the algebra A(D). This stands in contrast to the curve case: J. Bruna and J. M. Ortega have shown [Math. Ann. 274, 527-575 (1986; Zbl 0585.32018)] that smooth simple closed curves that are nowhere complex-tangential are nearly pluriharmonic interpolation sets in that if K is such a curve, then C(K) contains a subspace of finite codimension each element of which can be interpolated by a function pluriharmonic on D and continuous on the closure of D.

Reviewer: E.L.Stout

MSC:

32A38	Algebras of holomorphic functions of several complex variables
32T99	Pseudoconvex domains
31C10	Pluriharmonic and plurisubharmonic functions

Keywords:

pluriharmonic interpolation sets; peak-interpolation sets; algebra of holomorphic functions