. Let $X$ be a compact Kähler manifold of complex dimension $n$, let $\Omega>0$ be a smooth volume form, let $\gamma$ be a big closed positive $(1,1)$-current with continuous local potentials. Let also $\psi\in{\cal P}_{\gamma}\cap L^{\infty}(X)$ be a solution of the degenerate complex Monge-Ampère equation $(\gamma+i\partial\bar{\partial}\psi)^{n}=f\,\Omega\,,$ with $f\in L\log^{n+\varepsilon_{0}}L(X)$ for some $\varepsilon_{0}>0$. Then the following conclusions hold.
(A) There exist a uniform constant $C_{1}=C_{1}(\varepsilon_{0},\gamma,\Omega)>0$ such that for all $\varepsilon\in(0,\varepsilon_{0}]$ holds an estimate

$$\operatorname{Osc}(\psi)\leq C_{1}^{\,2n\left(1+\frac{n}{\varepsilon}\right)}\,\left[e\left(3+\frac{2n}{\varepsilon}\right)\right]^{\,\frac{\;n^{2}}{\varepsilon}}I_{\varepsilon}(f)^{\frac{n}{\varepsilon}}+\;1\,,$$

where

$$I_{\varepsilon}(f):=\{\gamma\}^{-n}\int\limits_{X}f\log^{n+\varepsilon}\left(e+\{\gamma\}^{-n}f\right)\Omega\,.$$

(B) Assume that the solution $\psi$ is continuous, normalized by the condition $\sup_{X}\psi=0$ and consider also a continuous solution $\varphi\in{\cal P}_{\gamma}$, $\sup_{X}\varphi=0$ of the degenerate complex Monge-Ampère equation $(\gamma+i\partial\bar{\partial}\varphi)^{n}=g\,\Omega\,,$ with $g\in L\log^{n+\varepsilon_{0}}L(X)$. Let $K>0$ be a constant such that $I_{\varepsilon_{0}}(f),I_{\varepsilon_{0}}(g)\leq K$. Then there exists a constant $C_{2}=C_{2}(\varepsilon_{0},\gamma,\Omega,K)>0$ such that

	$\displaystyle\|\varphi-\psi\|_{{}_{C^{0}(X)}}$	$\displaystyle\leq$	$\displaystyle 2C_{2}^{{}^{\alpha_{0}}}\,\left(\log\|\varphi-\psi\|^{-1}_{{}_{L^{1}(X,\,\Omega)}}\right)^{{}^{-\alpha_{0}}}\,,$
	$\displaystyle\alpha_{0}$	$\displaystyle:=$	$\displaystyle\frac{1}{(n+1+n^{2}/\varepsilon_{0})}\,,$

provided that the inequality $\|\varphi-\psi\|_{{}_{L^{1}(X,\,\Omega)}}\leq\min\{1/2,e^{-C_{2}}\}$ holds.
(C) Let $(\gamma_{t})_{t>0}$ be a family of currents satisfying the same properties as $\gamma$, fix a finite covering $(U_{\alpha})_{\alpha}$ of coordinate starshaped open sets, and let us write $\gamma_{t}=i\partial\bar{\partial}h_{t,\alpha}$ with $\sup_{U_{\alpha}}h_{t,\alpha}=0$ over $U_{\alpha}$ and $C_{1,t}:=C_{1}(\varepsilon_{0},\gamma_{t},\Omega)$, $C_{2,t}=C_{2}(\varepsilon_{0},\gamma_{t},\Omega,K)$. Assume
(C1) $\sup_{t>0}\max_{\alpha}\|h_{t,\alpha}\|_{L^{\infty}(U_{\alpha})}<+\infty$ and
(C2a) there exist a decomposition of the type $\gamma_{t}=\theta_{t}+i\partial\bar{\partial}u_{t}$, whith $\theta_{t}$ smooth, $\min_{X}u_{t}=0$, $\sup_{t>0}\max_{X}u_{t}<+\infty$ and $\theta_{t}\leq(\{\gamma_{t}\}^{n})^{1/n}\omega$ for some Kähler metric $\omega>0$ on $X$,
or
(C2b) the distributions $\gamma^{n}_{t}/\Omega$ are represented by functions and

$$\sup_{t>0}\;\;\{\gamma_{t}\}^{-n}\int\limits_{X}\log\left(e+\{\gamma_{t}\}^{-n}\gamma^{n}_{t}/\Omega\right)\gamma^{n}_{t}<+\infty\,.$$

Then $\sup_{t>0}C_{j,t}<+\infty$ for $j=1,2$.