A Bessel $\delta$-method and hybrid bounds for $\mathrm{GL}_2$

Let $g$ be a primitive holomorphic or Maass newform for $\Gamma_0(D)$. In this paper, by studying the Bessel integrals associated to $g$, we prove an asymptotic Bessel $\delta$-identity associated to $g$. Among other applications, we prove the following hybrid subconvexity bound $$L\left(1/2+it,g\otimes \chi\right)\ll_{g,\varepsilon} (q(1+|t|))^{\varepsilon}q^{3/8}(1+|t|)^{1/3}$$ for any $\varepsilon>0$, where $\chi \bmod q$ is a primitive Dirichlet character with $(q, D)=1$. This improves the previous known result.