Test vector conjecture for local triple product periods

Let $F=\mathbb{F}_v$, $G=\mathrm{PGL}_2(F)$, and let $\pi_1,\pi_2,\pi_3$ be unitary representations of $G$. Assume that $\pi_3$ is fixed and that $C(\pi_2)\geq C(\pi_3)=M$. For a representation over an archimedean field, let $\mathcal{S}_d$ denote its Sobolev norm of order $d$; for a non-archimedean field, let $K(M^B)$ denote the relevant congruence subgroup. The local triple product integral is denoted by $I^T$.

Test vector conjecture for local triple product periods. There exist unit vectors $\varphi_i\in\pi_i$ such that: (i) the Sobolev norms of $\varphi_3$ are controlled by $M$, namely, for some absolute constant $B$, $\varphi_3$ is $K(M^B)$-invariant when $F$ is non-archimedean, while for archimedean $F$ there is a constant $B(d)$ depending only on $d$ such that $\mathcal{S}_d(\varphi_3)\ll_d M^{B(d)}$; (ii) for some constant $A$,

$$I^T(\varphi_1,\varphi_2,\varphi_3)\gg_{\epsilon} C(\pi_1\otimes\pi_2\otimes\pi_3)^{-\frac14-\epsilon}M^A;$$

(iii) there exists $A'$ such that, for every $\psi\in\pi'$ with Sobolev norms controlled by $M$,

$$I^T(\varphi_2,\varphi_2,\psi)\ll_{\epsilon}M^{A'}C(\pi_1\otimes\pi_2\otimes\pi_3)^{-\frac14+\epsilon}\left(\frac{C(\pi_1\otimes\pi_2\otimes\pi_3)^{\frac12}}{C(\pi_2\otimes\pi_2)}\right)^{\vartheta}.$$

Here $\vartheta<\frac12$ is any bound towards the Ramanujan conjecture.

These vectors are intended for the period method for subconvexity: the triple product period formula reduces global estimates to lower bounds for local triple product integrals and upper bounds for related local periods. The conjecture is presented as the local input needed in the cited period-method framework; its resolution is not established in the supplied text.