Hybrid subconvexity for Maass form symmetric-square $L$-functions

Recently R. Khan and M. Young proved a mean Lindelöf estimate for the second moment of Maass form symmetric-square $L$-functions $L({\text{sym}}^2{u}_j,1\mathrm{/}2+it)$ on the short interval of length $G\gg \mathrm{\mid }{t}_j{\mathrm{\mid }}^{1+\epsilon }\mathrm{/}{t}^{2\mathrm{/}3}$, where $t_j$ is a spectral parameter of the corresponding Maass form. Their estimate yields a subconvexity estimate for $L({\text{sym}}^2{u}_j,1\mathrm{/}2+it)$ as long as $\mathrm{\mid }{t}_j{\mathrm{\mid }}^{6\mathrm{/}7+\delta }\ll t<(2-\delta )\mathrm{\mid }{t}_j\mathrm{\mid }$. We obtain a mean Lindelöf estimate for the same moment in shorter intervals, namely for $G\gg \mathrm{\mid }{t}_j{\mathrm{\mid }}^{1+\epsilon }\mathrm{/}t$. As a corollary, we prove a subconvexity estimate for $L({\text{sym}}^2{u}_j,1\mathrm{/}2+it)$ on the interval $\mathrm{\mid }{t}_j{\mathrm{\mid }}^{2\mathrm{/}3+\delta }\ll t\ll \mathrm{\mid }{t}_j{\mathrm{\mid }}^{6\mathrm{/}7-\delta }$.