Big o notation for sublinear algorithm in streaming algorithm
A function $f$ is $\log^{O(1)}N$ if there exists a constant $k$ such that $f(n) \leq \left(\log N\right)^k$ (and not $\underbrace{\log … \log}_{k\text{ times}} N$), so you were right on this point.
The notation $N^{\alpha}$, $\alpha &lt; 1$ is NOT another notation for $\log^{O(1)}N$ but another complexity family. If you chose $0 &lt; \alpha &lt; 1$ and $0 &lt; k$, $\log ^kN$ will always be $o(N^{\alpha})$.