On the $$\bar \partial $$ equation in weightedL 2 norms in ℂ1equation in weightedL 2 norms in ℂ1

Abstract

For a large class of subharmonicφ, the equation\(\bar \partial u = f\) is studied in\(\mathcal{H} = L^2 (\mathbb{C}^1 ,e^{ - \varphi } )\). Pointwise upper bounds are derived for the distribution kernels of the canonical solution operator and of the orthogonal projection onto the space of entire functions inH. Existence theorems inLp norms are derived as a corollary. A class of counterexamples, related to the failure of\(\bar \partial _b \) to be analytic-hypoelliptic on certain CR manifolds, is discussed.