Orthogonal projections onto subspaces of the harmonic Bergman space (Q2266146)

Let \(\Omega \subset {\mathbb{R}}^ m\) be a bounded, smooth domain. We construct a continuous linear operator \(T: W^ 0(\Omega)\to W^ 0(\Omega)\) which for all \(k\in ({\mathbb{N}}\cup \{\infty \})\) is actually continuous from \(W^ k(\Omega)\to W^ k_ 0(\Omega)\), and which moreover has the property that \(ST=S\), for any orthogonal projection S of \(W^ 0(\Omega)\) onto a subspace of the harmonic Bergman space. That is, the operator assigns to each function a function vanishing to high (infinite if \(k=\infty)\) order at \(b\Omega\), but with the same projection. S can in particular be the harmonic Bergman projection, or, when \(\Omega \subset {\mathbb{C}}^ n\), 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.