Let Ω ⊂Rm be a bounded,
smooth domain. We construct a continuous linear operator T : W0(Ω) → W0(Ω)
which for all k ∈ (N ∪{∞}) is actually continuous from Wk(Ω) → W0k(Ω), and
which moreover has the property that ST = S, for any orthogonal projection S of
W0(Ω) onto a subspace of the harmonic Bergman space. That is, the operator assigns
to each function a function vanishing to high (infinite if k = ∞) order at bΩ„ but
with the same projection. S can in particular be the harmonic Bergman projection,
or, when Ω ⊂ Cn, the (analytic) Bergman projection. The question whether such an
operator exists arises for example in connection with regularity properties of the
Bergman projection and their intimate connection with boundary regularity of
holomorphic mappings.
