Before revision:  In particular, for any real p > 2, we first obtain an algorithm for F_p moment estimation using \frac{1{\varepsilon^{4/p}}\cdot n^{1-2/p}} bits of memory. Our techniques also give algorithms for F_p moment estimationwith p>2 on arbitrary order insertion-only and turnstile streams, using \frac{1{\varepsilon^{4/p}}\cdot n^{1-2/p}} bits of space and two passes, which is the first optimal multi-pass F_p estimation algorithm up to \log n factors.
After revision:  In particular, for any real p > 2, we first obtain an algorithm for F_p moment estimation using \frac{1{\varepsilon^{4/p}}\cdot n^{1-2/p}}  \frac{1{\varepsilon^{4/p}}\cdot n^{1-2/p}} bits of space and two passes, which is the first optimal multi-pass F_p estimation algorithm up to \log n factors.