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

Let $g$ be a primitive holomorphic or Maass newform for $Γ_0(D)$. In this paper, by studying the Bessel integrals associated to $g$, we prove an asymptotic Bessel $δ$-identity associated to $g$. Among other applications, we prove the following hybrid subconvexity bound $L\left(1/2+it,g\otimes χ\right)\ll_{g,\varepsilon} (q(1+|t|))^{\varepsilon}q^{3/8}(1+|t|)^{1/3}$ for any $\varepsilon>0$, where $χ\bmod q$ is a primitive Dirichlet character with $(q, D)=1$. This improves the previous known result.

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