  Let $(X,\omega)$ be a compact K\"ahler manifold. We obtain uniform H\"older
regularity for solutions to the complex Monge-Amp\`ere equation on $X$ with
$L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the
ample locus in any big cohomology class. We also study the range
$\MAH(X,\omega)$ of the complex Monge-Amp\`ere operator acting on
$\omega$-plurisubharmonic H\"older continuous functions. We show that this set
is convex, by sharpening Ko{\l}odziej's result that measures with $L^p$-density
belong to $\MAH(X,\omega)$ and proving that $\MAH(X,\omega)$ has the
"$L^p$-property", $p>1$. We also describe accurately the symmetric measures it
contains.
