In this paper, we use the Sobolev type inequality in Wang et al. (Moser–Trudinger inequality for the complex Monge–Ampère equation, arXiv: 2003.06056v1 (2020)) to establish the uniform estimate and the Hölder continuity for solutions to the complex Monge–Ampère equation with the right-hand side in $L^p$ for any given $p>1$. Our proof uses various PDE techniques but not the pluripotential theory.