Weyl Bounds for ${\mathrm{GL}}_2\times {\mathrm{GL}}_2$ Rankin--Selberg $L$-functions via Short Spectral Moments

Let $f$ and $g$ be Hecke–Maass cusp forms of ${\mathrm{SL}}_2(\mathbb{Z})$, with spectral parameters $t_f$ and $t_g$, respectively. Let $T$ be a large parameter and put $L=\mathrm{\mid }T-t_g\mathrm{\mid }\mathrm{.}$ We prove the Lindelöf-on-average estimate $$\sum _{\mathrm{\mid }{t}_f-T\mathrm{\mid }\le H}{\mid L\text{\hspace{0.17em}⁣}(\frac{1}{2},f\otimes g)\mid }^2{\ll }_{\epsilon }{T}^{1+\epsilon }H,$$ uniformly in the range $$T^{1\mathrm{/}3+\epsilon }\ll H,H{T}^{\epsilon }\ll L\le T\mathrm{.}$$ As an immediate consequence, in the generic range when both $\mathrm{\mid }{t}_g\mathrm{\mid }$ and $L$ are of size $T$, we obtain the Weyl bound $$L\text{\hspace{0.17em}⁣}(\frac{1}{2},f\otimes g){\ll }_{\epsilon }C(f\times g{)}^{1\mathrm{/}6+\epsilon }\mathrm{.}$$