Longest palindrome substring in logarithmic runtime complexity
If a you are given a palindrome $p$ of size $N$ (as you say in the beginning of the question), then the longest palindrome is $p$ itself and you don't need to do any computation.

If your input is not a palindrome but an arbitrary string, then you need to at least read the input and therefore $\Omega(N)$ is a trivial lower bound.