On $p$-adic Asai $L$-functions of Bianchi modular forms at non-ordinary primes and their decomposition into bounded $p$-adic $L$-functions

Let $p$ be an odd prime integer, $F\mathrm{/}\mathbb{Q}$ be an imaginary quadratic field, and $\mathrm{\Psi }$ be a small slope cuspidal Bianchi modular form over $F$ which is non-ordinary at $p$. In this article, we first construct a $p$-adic distribution $L_p^{\mathrm{A}\mathrm{s}}(\mathrm{\Psi })$ that interpolates the twisted critical $L$-values of Asai (or twisted tensor) $L$-function of $\mathrm{\Psi }$, generalizing the works of Loeffler–Williams from the ordinary case to the non-ordinary case. To obtain this distribution, we construct some polynomials using Asai–Eisenstein elements: the Betti analogue of the Euler system machinery, developed by Loeffler–Williams. We use some techniques analogous to those of Loeffler–Zerbes for interpolating the twists of Beilinson–Flach elements arising in the Euler system associated with Rankin–Selberg convolutions of elliptic modular forms. We also use the interpolation method developed by Amice–Vélu, Perrin-Riou, and Büyükboduk–Lei in the construction. Furthermore, under some assumptions, we decompose these unbounded $p$-adic distributions into the linear combination of bounded measures as done by Pollack, Sprung, and Lei–Loeffler–Zerbes in the elliptic modular forms case.