Weighted and boundary l p estimates for solutions of the $\partial$ -equation on lineally convex domains of finite type and applications

Abstract

We obtain sharp weighted estimates for solutions of the equation $\partial$ 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 ($\partial$ $Ω$) and BMO($\partial$ $Ω$) 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.