Signed $p$-adic $L$-functions of Bianchi modular forms

Let $p\ge 3$ be a prime number and $K$ be a quadratic imaginary field in which $p$ splits as $\mathfrak{p}\bar{\mathfrak{p}}$. Let $\mathcal{F}$ be a cuspidal Bianchi eigenform over $K$ of weight $(k,k)$, where $k\ge 0$ is an integer, level $\mathfrak{m}$ coprime to $p$, and non-ordinary at both of the primes above $p$. We assume $\mathcal{F}$ has trivial nebentypus. For $\mathfrak{q}\in \{\mathfrak{p},\bar{\mathfrak{p}}\}$, let $a_{\mathfrak{q}}$ be the $T_{\mathfrak{q}}$ Hecke eigenvalue of $\mathcal{F}$ and let ${\alpha }_{\mathfrak{q}},{\beta }_{\mathfrak{q}}$ be the roots of polynomial $X^2-a_{\mathfrak{q}}X+p^{k+1}$. Then we have four $p$-stabilizations of $\mathcal{F}$: ${\mathcal{F}}^{{\alpha }_{\mathfrak{p}},{\alpha }_{\bar{\mathfrak{p}}}},{\mathcal{F}}^{{\alpha }_{\mathfrak{p}},{\beta }_{\bar{\mathfrak{p}}}},{\mathcal{F}}^{{\beta }_{\mathfrak{p}},{\alpha }_{\bar{\mathfrak{p}}}},$ and ${\mathcal{F}}^{{\beta }_{\mathfrak{p}},{\beta }_{\bar{\mathfrak{p}}}}$ which are Bianchi cuspforms of level $p\mathfrak{m}$. By the works of Williams, to each $p$-stabilization ${\mathcal{F}}^{\ast ,†}$, we can attach a locally analytic distribution $L_p({\mathcal{F}}^{\ast ,†})$ over the ray class group $\text{Cl}(K,p^{\mathrm{\infty }})$. On viewing $L_p({\mathcal{F}}^{\ast ,†})$ as a two-variable power series with coefficients in some $p$-adic field having unbounded denominators satisfying certain growth conditions, we decompose this power series into a linear combination of power series with bounded coefficients in the spirit of Pollack, Sprung, and Lei–Loeffler–Zerbes.