Sobolev regularity of the Bergman and Szegö projections in terms of ∂⊕∂^* and ∂_b⊕∂_b^*

2024/10/13 by Emil J. Sträube, Straube, Emil J.
Mathematics · #32W05 #32W10 #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2410.09996

openalex publication_date 2024/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be a smooth bounded pseudoconvex domain in ℂⁿ. It is shown that for 0≤ q≤ n, s≥ 0, the embedding j_q: dom(∂)∩ dom(∂^*) \hookrightarrow L²_(0,q)(Ω) is continuous in W^s(Ω)--norms if and only if the Bergman projection P_q is (see below for the modification needed for j₀). The analogous result for the operators on the boundary is also proved (for n≥ 3). In particular, j₁ is always regular in Sobolev norms in ℂ², notwithstanding the fact that N₁ need not be.

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