On the Second Moment of $L(1\mathrm{/}2,\mathrm{A}\mathrm{s}(f)\times \phi )$

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 $\mathrm{A}\mathrm{s}(f)$ 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 $L(1\mathrm{/}2,\mathrm{A}\mathrm{s}(f)\times \phi )$ in the $k$-aspect.