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

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 $X\mathrm{.}$ 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 $(d{d}^c\phi +\omega {)}^n=\mu$ has a H"older continuous solution.