Complex Monge-Ampère equation for measures supported on real submanifolds

Let $(X,ω)$ be a compact $n$-dimensional Kähler manifold on which the integral of $ω^n$ 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 $μ$ be a probability measure compactly supported on $K$ with $L^p$ density for some $p>1.$ We prove that the complex Monge-Ampère equation $(dd^c φ+ ω)^n=μ$ has a Hölder continuous solution.

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