Finite entropy vs finite energy
The authors show that the solutions to the degenerate complex Monge-Ampère equation, with the right hand side being a probability measure having bounded entropy, belong to a certain finite energy space. More exactly, let \((X, \omega)\) be a Kähler manifold with volume one and complex dimension \(n\). The authors shows in Theorem A that the solutions \(\phi\) to the equation \[ \omega_\phi^n := (\omega + dd^c \phi)^n = f \omega^n, \] where the probability measure \(f \omega^n\) is assumed to have finite entropy \[ \int_{X} f \log f \omega^n, \] belong to the finite energy class \(\mathcal{E}^p_\omega\) for \(p = \frac{n}{n-1}\) and \(n>1\). The latter space consists of those elements of \(\mathrm{PSH}(X,\omega)\) which satisfy \[ E_p(\phi):=\int_X |\phi|^p \omega_\phi^n < \infty. \] What is more, it is shown that the bound on \(p\) is optimal in Theorem A. On the road for this result the authors provide another interesting result, Theorem B, which is known as the complex Moser-Trudinger inequality. Namely, for the normalized elements \(\phi\) of \(\mathcal{E}^p_\omega\) the inequality \[ \int_X e^{(\mathrm{const} |E_p(\phi)|^{- \frac 1 n}|\phi|^{1+\frac p n})}\omega^n < \mathrm{const} \] is shown to hold.
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