When $Aut(X)$, the group of holomorphic automorphisms of $X$, acts transitively on $X$, one can regularize $\omega$-psh functions by averaging over the Haar measure of the connected component of the identity of $Aut(X)$. This is very similar to the way one regularizes psh functions in $\mathbb{C}^{n}$ by using convolutions with an approximation of the identity for the convolution product. We refer the reader to [Hu] and the Appendix of [G] for more details.

Let $\varphi_{h}$ be the $\omega$-psh function which is the translate of $\varphi$ by an automorphism which is at distance $h$ from identity. We use the notation $\varphi_{h}$ by analogy with the $\mathbb{C}^{n}$-situation, where $\varphi_{h}(x)=\varphi(x+h)$. Since $\varphi$ is bounded, it has gradient in $L^{2}$, hence

$$||\varphi_{h}-\varphi||_{L^{2}}\leq C|h|,$$

by using Cauchy-Schwarz inequality in a local chart. We can thus apply proposition 3.2 to obtain that

$$||\varphi_{h}-\varphi||_{L^{\infty}}\leq C^{\prime}|h|^{\gamma},$$

for all $\gamma<2/(2+nq)$. Since $\varphi_{h}(x)\simeq\varphi(x+h)$ in a local chart, this precisely means that $\varphi$ is Hölder-continuous of exponent $\gamma$. ∎