Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

Tran Vu Khanh and Andrew Raich

Mathematische Nachrichten, 2021, vol. 294, issue 1, 82-97

Abstract: We prove Lp estimates, 1≤p≤∞, for solutions to the tangential Cauchy–Riemann equations ∂¯bu=ϕ on a class of infinite type domains Ω⊂C2. The domains under consideration are a class of convex ellipsoids, and we show that if ϕ is a ∂¯b‐closed (0,1)‐form with coefficients in Lp, then there exists an explicit solution u satisfying ∥u∥Lp(bΩ)≤C∥ϕ∥Lp(bΩ). Moreover, when p=∞, we show that there is a gain in regularity to an f‐Hölder space. We also present two applications. The first is a solution to the ∂¯‐equation, that is, given a smooth (0,1)‐form ϕ on bΩ with an L1‐boundary value, we can solve the Cauchy–Riemann equation ∂¯u=ϕ so that ∥u∥L1(bΩ)≤C∥ϕ∥L1(bΩ) where C is independent of u and ϕ. The second application is a discussion of the zero sets of holomorphic functions with zero sets of functions in the Nevanlinna class within our class of domains.

Date: 2021
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https://doi.org/10.1002/mana.201900282

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