On the Second Moment of $ L (1/2, \mathrm{As}(f) \times φ)$

Let $\mathbf{F} = \mathbf{Q}(\sqrt D)$ be a real quadratic field. In this paper, we establish a large sieve inequality for the Asai lifts $\mathrm{As} (f)$ with $f$ in a Hecke orthonormal basis of the space of Hilbert modular cusp forms of parallel weight $(k, k)$ over $\mathbf{F}$. As an application, for a fixed Hecke--Maass cusp form $\phi$ over $\mathbf{Q}$, we prove a non-trivial bound for the second moment of the convoluted central $L$-values $L (1/2, \mathrm{As}(f) \times \phi)$ in the $k$-aspect.