Sub-convexity problem for Rankin-Selberg $L$-functions

We establish a sub-convexity estimate for Rankin-Selberg $L$-functions in the combined level aspect, using the circle method. If $p$ and $q$ are distinct prime numbers, $f$ and $g$ are non-exceptional newforms (modular or Maass) for the congruence subgroups $\Gamma_0(p)$ and $\Gamma_0(q)$ (resp) with trivial nebentypus, then for all $\epsilon >0$ we show that there exists an $A >0$ such that $$L\left(\frac{1}{2}+it, f \times g \right) \ll_{\epsilon,\mu_f, \mu_g}(1+|t|)^A \frac{(pq)^{1/2+\epsilon}}{\max\{p,q \}^{\frac{1}{64}}}.$$ The dependence on $\mu_f$ and $\mu_g$, the parameters at infinity for $f$ and $g$ respectively, is polynomial. Further, if $p$ is fixed and $q \rightarrow \infty$, we improve this to $$L\left(\frac{1}{2}+it, f \times g \right) \ll_{\epsilon,\mu_f,\mu_g}(p(1+|t|))^Aq^{\frac{1}{2}-\frac{1-2\theta}{27+28\theta}+\epsilon} ,$$ where $\theta$ is the exponent towards Ramanujan-conjecture for cuspidal automorphic forms. Unconditionally, we can take $\theta = 7/64$. This improves all previously known sub-convexity estimates in this case.