Moments of Hardy's function over short intervals

Let as usual $Z(t)=\zeta (1\mathrm{/}2+it){\chi }^{-1\mathrm{/}2}(1\mathrm{/}2+it)$ denote Hardy’s function, where $\zeta (s)=\chi (s)\zeta (1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z^{\mathrm{\prime }}(t)$ are proved. It is also proved that $$H(\log T{)}^{k^2}{\ll }_{k,\alpha }\sum _{T<\gamma \le T+H}\underset{\gamma \le {\tau }_{\gamma }\le {\gamma }^{+}}{\max }\mathrm{\mid }\zeta (1\mathrm{/}2+i{\tau }_{\gamma }){\mathrm{\mid }}^{2k}{\ll }_{k,\alpha }H(\log T{)}^{k^2}\mathrm{.}$$ Here $k>1$ is a fixed integer, $\gamma ,{\gamma }^{+}$ denote ordinates of consecutive complex zeros of $\zeta (s)$ and $T^{\alpha }\le H\le T$, where $\alpha$ is a fixed constant such that $0<\alpha \le 1$. This sharpens and generalizes a result of M.B. Milinovich.