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We obtain sharp weighted estimates for solutions of the equation $\overline{\partial }u=f$ in a lineally convex domain of finite type. Precisely we obtain estimates in the spaces $L^{p}(\Omega ,\delta ^{\gamma })$, $\delta $ 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) $, $\gamma >-1$, we have a mixed gain on the index p and the exponent $\gamma $; secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic $L^{p}$ estimate with weight $\delta _{\Omega }^{\gamma +1}$. Moreover we extend those results to $\gamma =-1$ and obtain $L^{p}(\partial \Omega )$ and $BMO(\partial \Omega )$ estimates. These results allow us to extend the $L^{p}(\Omega ,\delta ^{\gamma })$-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.

mathematische zeitschrift

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