In this paper, we investigate the stability and instability of standing waves for the Schrödinger-Choquard equation with mixed fractional Laplacians. We first establish the existence and stability of normalized standing waves in the $L^2$-subcritical case. Subsequently, we prove the existence and strong instability of normalized ground state standing waves in the $L^2$-supercritical case.

In this paper, we established the boundedness of higher-order commutators $I_{\beta ,b}^m$ generated by the fractional integral operator with BMO functions on grand weighted variable-exponent Herz-Morrey spaces ${\mathrm{M}\dot{\mathrm{K}}}_{\lambda ,p(\cdot )}^{\alpha ,r),\theta }(\omega )$. We also obtained the boundedness of the $m-$order multilinear fractional Hardy operator ${\mathcal{H}}_{\beta ,m}$ and its adjoint operator ${\mathcal{H}}_{\beta ,m}^{\ast }$ on weighted variable-exponent Herz-Morrey spaces ${\mathrm{M}\dot{\mathrm{K}}}_{q,p(\cdot )}^{\alpha ,\lambda }(\omega )$.