How to solve $T(n)= 4T(\sqrt n) +\log^2n$?
Let $S(n) = T(2^n)$. Then
$$
S(n) = T(2^n) = 4T(2^{n/2}) + n^2 = 4S(n/2) + n^2.
$$
You can solve this recurrence using the master theorem, and then use $T(n) = S(\log n)$ to obtain a solution for the original recurrence.