Hardy spaces and integral formulas for 𝒟ℱ𝒩‐domains with arbitrary boundary

Let E be the dual of a Fréchet nuclear space, then it is well‐known that for each open set U in E the space ℋ( U ) of all holomorphic functions on U is a nuclear Fréchet space. Let be a commutative unital Banach sub‐algebra of all bounded holomorphic functions on U which separates points. Applying the nuclearity of ℋ( U ) we show that the evaluation on U is given by an integral formula over the Shilov boundary of . We obtain Szegö‐ and Bergman kernels together with some boundary estimates. Moreover, we show that there is a notion of Hardy and Bergman space for ℱ‐domains with arbitrary boundary. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)