Estimates $ L^{r}-L^{s}$ for solutions of the $\bar \partial $ equation in strictly pseudo convex domains in ${\mathbb{C}}^{n}.$

Eric Amar (1)

1 IMB - Institut de Mathématiques de Bordeaux

We reprove estimates for solutions of the $\bar \partial u=\omega $ equation in a strictly pseudo convex domain $\displaystyle \Omega $ in ${\mathbb{C}}^{n}.$ For instance if the $\displaystyle (p,q)$ current $\omega $ has its coefficients in $\displaystyle L^{r}(\Omega )$ with $\displaystyle 1\leq r<2(n+1)$ then there is a solution $u$ in $\displaystyle L^{s}(\Omega )$ with $\displaystyle \ \frac{1}{s}=\frac{1}{r}-\frac{1}{2(n+1)}.$ These results were already done by S. Krantz~\cite{KrantzDbar76} and we propose an other approach based on Carleson measures of order $\alpha $ introduced and studied in~\cite{AmarBonami} and on the subordination lemma~\cite{subPrinAmar12}.

Mots clés: Carleson measures; d_bar equation

Domaines: Variables complexes [math.CV]

Eric Amar. Estimates $ L^{r}-L^{s}$ for solutions of the $\bar \partial $ equation in strictly pseudo convex domains in ${\mathbb{C}}^{n}.$. 2013. ⟨hal-00922356v2⟩