Multipliers and weighted \(\bar\partial\)-estimates. (Q1809938)

Let \(\Omega \) be a domain in \(\mathbb C.\) A measure \(\mu \) in \(\Omega \) is called a locally doubling measure whenever there is a constant \(C>1\) such that \(\mu (B)\leq C \mu (B'),\) for all balls \(B\subset \Omega\) of radius smaller than \(1,\) where \(B\) is the ball with the same center as \(B'\) and twice its radius. The main result of this paper is the following: Let \(\phi \) be a subharmonic function in the unit disk \(\mathbb D\) such that \(\Delta \phi (D(z,r)) >1\) for some \(r>0,\) where \(D(z,r)\) is any hyperbolic disk with center \(z\in \mathbb D\) and radius \(r.\) Moreover it is assumed that \(\Delta \phi\) is a locally doubling measure with respect to hyperbolic distance. Then there is a solution \(u\) to the equation \(\overline \partial u=f\) with \[ \int_{\mathbb D} \frac{| u(z)| ^p}{1-| z| ^2}\, e^{-\phi }\, dm(z) \lesssim \int_{\mathbb D} \frac{| f(z) (1-| z| ^2)^p}{1-| z| ^2}\, e^{-\phi }\, dm(z), \] for any \(p\in [1,+\infty )\) and \(\sup | u| \, e^{-\phi } \lesssim \sup | f(\zeta ) (1-| \zeta | )\, e^{-\phi (\zeta )}.\) Here the measure \(\Delta \phi \) is only locally doubling. The proof uses the construction of the so-called multipliers. These are holomorphic functions that have very precise growth control.