Suppose we compute the correlation PCA of a dataset $X$ (with $m$ variables and $n$ observations) by first normalizing the input variables. That is: mean $\rightarrow 0$ and standard deviation $\rightarrow 1$. Let us assume for the sake of this question that $\mu_i=0$ for our dataset. In that case we only need to normalize the standard deviation:
$$X'_{i,j}={X_{i,j}\over \sigma_i}$$
Once the correlation matrix $X'X'^T$ is computed, we calculate its SVD which provides us with the eigenvectors $U$.
To rotate/transform the input points in accordance with the eigenvectors we multiply them with $U^T$. My question now is do we perform this on the original input samples ($X$) or on the normalized samples ($X'$) ?
If you're using the SVD, you apply it to the original data matrix to get the eigenvectors of the correlation matrix.
The transformation given by PCA is applied to the standardized samples (i.e. those in $X'$).