LAUSR

We obtain sharp weighted estimates for solutions of the equation $$\overline{\partial }u=f$$∂¯u=f in a lineally convex domain of finite type. Precisely we obtain estimates in the spaces $$L^{p}(\Omega ,\delta ^{\gamma })$$Lp(Ω,δγ), $$\delta $$δ being the distance to the boundary, with two different types of hypothesis on the form f: first, if the data f belongs to $$L^{p}\left( \Omega ,\delta _{\Omega }^{\gamma }\right) $$LpΩ,δΩγ, $$\gamma >-1$$γ>-1, we have a mixed gain on the index p and the exponent $$\gamma $$γ; secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic $$L^{p}$$Lp estimate with weight $$\delta _{\Omega }^{\gamma +1}$$δΩγ+1. Moreover we extend those results to $$\gamma =-1$$γ=-1 and obtain $$L^{p}(\partial \Omega )$$Lp(∂Ω) and $$BMO(\partial \Omega )$$BMO(∂Ω) estimates. These results allow us to extend the $$L^{p}(\Omega ,\delta ^{\gamma })$$Lp(Ω,δγ)-regularity results for weighted Bergman projection obtained in Charpentier et al. (Complex Var Elliptic Equ 59(8):1070–1095, 2014) for convex domains to more general weights.