Håkan Samuelsson

Uniform algebras and approximation on manifolds

Let Ω⊂ℂn be a bounded domain and let

OPERATORS WITH SMOOTH FUNCTIONAL CALCULI

\mathcal{A} \subset\mathcal{C}(\overline{\Omega})

A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas

be a uniform algebra generated by a set F of holomorphic and pluriharmonic functions. Under natural assumptions on Ω and F we show that the only obstruction to

Weighted Koppelman formulas and the ∂¯-equation on an analytic space

\mathcal{A} = \mathcal {C}(\overline{\Omega})

Regularizations of residue currents

is that there is a holomorphic disk

On the Briancon-Skoda theorem on a singular variety

D \subset\overline{\Omega}

Regularizations of Products of Residue and Principal Value Currents

such that all functions in F are holomorphic on D, i.e., the obvious obstruction is the only one. This generalizes work by A. Izzo. We also have a generalization of Wermer’s maximality theorem to the (distinguished boundary of the) bidisk.

Koppelman formulas and the

We introduce a class of (tuples of commuting) unbounded operators on a Banach space, admitting smooth functional calculi, which contains all operators of Helffer-Sjöstrand type and is closed under the action of smooth proper mappings. Moreover, the class is closed under tensor product of commuting operators. In general, and operator in this class has no resolvent in the usual sense, so the spectrum must be defined in terms of the functional calculus. We also consider invariant subspaces and spectral decompositions.