Bilinear Strichartz estimate conjecture for compact symmetric spaces

Let $M$ be a symmetric space of compact type of dimension $d$ and rank $r$, and let $M_0$ range over its irreducible components, with dimension $d_0$ and rank $r_0$. For $\lambda_1,\lambda_2 ef\Lambda^+$, let $P_{\lambda_1}$ and $P_{\lambda_2}$ denote the joint spectral projections associated with these parameters. Bilinear Strichartz estimate conjecture. (i) If $d_0\geq 3r_0$ for every irreducible component $M_0$, then for every $f,g\in L^2(M)$ and $\lambda_1,\lambda_2\in\Lambda^+$,

$$\|P_{\lambda_1}f\cdot P_{\lambda_2}g\|_{L^2(M)}\lesssim_{\varepsilon}(\min(|\lambda_1|,|\lambda_2|)+1)^{\frac{d}{2}-r+\varepsilon}\|f\|_{L^2(M)}\|g\|_{L^2(M)}.$$

(ii) If the stronger condition $d_0>3r_0$ holds for every irreducible component $M_0$, then the same estimate holds without the $\varepsilon$-loss. This conjectural estimate is proposed as the key replacement for bilinear estimates for Laplace–Beltrami eigenfunctions on compact symmetric spaces of higher rank, with applications to local well-posedness for cubic nonlinear Schrödinger equations. The paper presents it as an aim of the discussion rather than an established result, and no resolution is supplied here.