Let \(p\geqslant 3\) be a prime number and K be a quadratic imaginary field in which p splits as \(\mathfrak {p}\overline{\mathfrak {p}}\) . Let \(\mathcal {F}\) be a cuspidal Bianchi eigenform over K of weight (k, k), where \(k\geqslant 0\) 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 \(\mathfrak {q}\in \{\mathfrak {p}, \overline{\mathfrak {p}}\}\) , 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 \(\mathcal {F}\) : \(\mathcal {F}^{\alpha _{\mathfrak {p}},\alpha _{\overline{\mathfrak {p}}}}, \mathcal {F}^{\alpha _{\mathfrak {p}},\beta _{\overline{\mathfrak {p}}}}, \mathcal {F}^{\beta _{\mathfrak {p}},\alpha _{\overline{\mathfrak {p}}}},\) and \( \mathcal {F}^{\beta _{\mathfrak {p}},\beta _{\overline{\mathfrak {p}}}}\) which are Bianchi cuspforms of level \(p\mathfrak {m}\) . By the works of Williams, to each p-stabilization \(\mathcal {F}^{*,\dagger }\) , we can attach a locally analytic distribution \(L_{p}(\mathcal {F}^{*,\dagger })\) over the ray class group \(\text {Cl}(K,p^{\infty })\) . On viewing \(L_{p}(\mathcal {F}^{*,\dagger })\) 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.
