Complex Monge-Amp\`ere equation for measures supported on real submanifolds

Abstract

Let $(X,\omega)$ be a compact $n$-dimensional K\"ahler manifold on which the integral of $\omega^n$ is $1$. Let $K$ be an immersed real $\mathcal{C}^3$ submanifold of $X$ such that the tangent space at any point of $K$ is not contained in any complex hyperplane of the (real) tangent space at that point of $X.$ Let $\mu$ be a probability measure compactly supported on $K$ with $L^p$ density for some $p>1.$ We prove that the complex Monge-Amp\`ere equation $(dd^c \varphi + \omega)^n=\mu$ has a H\"older continuous solution.