On the moments of the Riemann zeta-function in short intervals

Assuming the Riemann Hypothesis it is proved that, for fixed $k>0$ and $H = T^\theta$ with fixed $0<\theta \le 1$, $$\int_T^{T+H}|\zeta(1/2+it)|^{2k} dt \ll H(\log T)^{k^2(1+O(1/\log_3T))},$$ where $\log_jT = \log(\log_{j-1}T)$. The proof is based on the recent method of K. Soundararajan for counting the occurrence of large values of $\log|\zeta(1/2+it)|$, who proved that $$\int_0^{T}|\zeta(1/2+it)|^{2k} dt \ll_\epsilon T(\log T)^{k^2+\epsilon}.$$