Hörmander's L2 estimates

Remarks

The proof is a duality argument in weighted L2 spaces. One views $\bar\partial$ as a closed, densely defined operator between $L^2$ spaces with weight $e^{-\varphi}$. The key step is the a priori inequality $\norm{\bar\partial^* f}^2 + \norm{\bar\partial f}^2 \ge \int c \abs{f}^2 e^{-\varphi}$ for forms $f$ in the domains of both operators. This Bochner–Kodaira–Morrey–Kohn type identity is where the plurisubharmonicity of $\varphi$ enters. The Riesz representation theorem then gives the solution $u$ together with the bound; pseudoconvexity is used to exhaust $\Omega$ by sets on which the argument applies. There is a version for any plurisubharmonic weight $\varphi$ on a pseudoconvex $\Omega$. For every $\bar\partial$-closed $(0,1)$-form $g$ with $\int \abs{g}^2 e^{-\varphi} < \infty$ there is $u$ with $\bar\partial u = g$ and $\int \abs{u}^2 e^{-\varphi} (1 + \abs{z}^2)^{-2} \le \int \abs{g}^2 e^{-\varphi}$. Similar statements hold for $(p,q)$-forms and on Stein and complete Kähler manifolds with Hermitian line bundles. Example: for $\varphi(z) = \abs{z}^2$ one may take $c = 1$, so every $\bar\partial$-closed $g$ with $\int \abs{g}^2 e^{-\abs{z}^2} < \infty$ is $\bar\partial u$ for some $u$ with $\int \abs{u}^2 e^{-\abs{z}^2} \le \int \abs{g}^2 e^{-\abs{z}^2}$. Pseudoconvexity cannot be dropped: an open set $\Omega \subseteq \C^n$ on which every smooth $\bar\partial$-closed $(0,q)$-form, $1 \le q \le n - 1$, is $\bar\partial$ of a smooth form is pseudoconvex.

Going deeper

For a student

The estimates turn the main existence problems of several complex variables into a single analytic tool. To build a holomorphic function with given properties, one first builds a smooth function with these properties. Then one corrects it by a solution of a $\bar\partial$-equation whose weighted $L^2$ norm is small. In this way one obtains Oka's solution of the Levi problem, the Cousin problems on pseudoconvex domains, and Cartan's theorem B for the structure sheaf. The method is the basis of the Ohsawa–Takegoshi extension theorem and of the theory of multiplier ideals.