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L p ‐Estimates for the ∂ ¯ b ‐equation on a class of infinite type domains
Author(s) -
Khanh Tran Vu,
Raich Andrew
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201900282
Subject(s) - mathematics , holomorphic function , mathematical analysis , class (philosophy) , pure mathematics , cauchy–riemann equations , type (biology) , zero (linguistics) , regular polygon , space (punctuation) , boundary (topology) , geometry , ecology , linguistics , philosophy , artificial intelligence , computer science , biology
We prove L p estimates, 1 ≤ p ≤ ∞ , for solutions to the tangential Cauchy–Riemann equations∂ ¯ b u = ϕ on a class of infinite type domains Ω ⊂ C 2 . The domains under consideration are a class of convex ellipsoids, and we show that ifϕ is a∂ ¯ b ‐closed (0,1)‐form with coefficients in L p , then there exists an explicit solution u satisfying∥ u ∥L p ( b Ω )≤ C ∥ ϕ ∥L p ( b Ω ). 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 b Ω with an L 1 ‐boundary value, we can solve the Cauchy–Riemann equation∂ ¯ u = ϕ so that∥ u ∥L 1 ( b Ω )≤ C ∥ ϕ ∥L 1 ( b Ω )where C is independent ofu 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.