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

Source abstract

Let $f$ and $g$ be Hecke--Maass cusp forms of $\operatorname{SL}_2(\mathbb{Z})$, with spectral parameters $t_f$ and $t_g$, respectively. Let $T$ be a large parameter and put $L = |T-t_g|.$ We prove the Lindelöf-on-average estimate $$\sum_{ |t_f-T|\leq H} \left|L\!\left(\frac12,f\otimes g\right)\right|^2 \ll_{\varepsilon} T^{1+\varepsilon}H,$$ uniformly in the range $$T^{1/3+\varepsilon}\ll H,\qquad HT^\varepsilon\ll L\leq T.$$ As an immediate consequence, in the generic range when both $|t_g|$ and $L$ are of size $T$, we obtain the Weyl bound $$L\!\left(\frac12,f\otimes g\right) \ll_\varepsilon C(f\times g)^{1/6+\varepsilon}.$$