Weighted and boundary $$L^{p}$$ L p estimates for solutions of the $$\overline{\partial }$$ ∂ ¯ -equation on lineally convex domains of finite type and applications

Abstract

We obtain sharp weighted estimates for solutions of the equation ∂ u = f in a lineally convex domain of finite type. Precisely we obtain estimates in the spaces L p (Ω,δ γ ), δ being the distance to the boundary, with two different types of hypothesis on the form f : first, if the data f belongs to L p Ω,δ γ Ω , γ > −1, we have a mixed gain on the index p and the exponent γ; secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic L p estimate with weight δ γ+1 Ω . Moreover we extend those results to γ = −1 and obtain L p (∂ Ω) and BMO(∂ Ω) estimates. These results allow us to extend the L p (Ω,δ γ )-regularity results for weighted Bergman projection obtained in [CDM14b] for convex domains to more general weights.2010 Mathematics Subject Classification. 32T25, 32T27.

Zeros Sets Of H^p Functions In Lineally Convex Domains Of Finite Type In C^n

“…ã the kernels K 1 ε and P ε are associated to the defining function ρ + Cε as described in [CDM14,CD]. We proceed as H Skoda in [Sko76, Section 8].…”

Sobolev and Hölder Estimates for Homotopy Operators of $\overline\partial$-Equations on Convex Domains of Finite Multitype

“…The Hölder estimate L ∞ → C 1 m was obtained [DFF99], the anisotropic version was later obtained by Fischer [Fis04] and Diederich-Fischer [DF06]. The L p -estimate was first done by Fischer and later there were partial progress by [AC03,Ahn04] and by [CD18]. The C k → C k+ 1 m estimate was achieved by Alexandre [Ale06].…”