Resumen de Convexité et equations de Cauchy-Riemann avec estimations Lp

Corinne Jouenne

• We obtain a solution of the equation $\overline{\partial} u=f$ as an integral supported only on the bounded convex domain D of $\mathbb{C}^n$, without finite type assumption. We show that u is in ${L}^p_{(0,q-1)}(D)$ if $f\in {L}^p_{(0,q)}(D)$ and $\frac{\overline{\partial}\rho\wedge f}{(-\rho)^{1-\varepsilon}}\in {L}^p_{(0,q+1)}(D)$ for some $\varepsilon$ depending on p and q, $0<\varepsilon\le 1$.