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

## TL;DR

This paper establishes sharp weighted estimates for solutions of the $ar{
abla}$-equation in lineally convex domains of finite type, extending regularity results to more general weights and boundary conditions.

## Contribution

It provides new weighted Lp and BMO estimates for solutions of the $ar{
abla}$-equation in lineally convex domains, including boundary and anisotropic cases, generalizing previous results.

## Key findings

- Sharp weighted estimates in Lp spaces with boundary distance weights.
- Extension of regularity results to more general weights and boundary conditions.
- New estimates for solutions with anisotropic data in convex domains.

## 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 ($\Omega$,$\delta$ $\gamma$), $\delta$ 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 $\Omega$,$\delta$ $\gamma$ $\Omega$ , $\gamma$ > --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 $\delta$ $\gamma$+1 $\Omega$. 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 [CDM14b] for convex domains to more general weights.