Unconditional Frames of Translates in $L_p(\mathbb{R}^d)$

We show that, for $1<p \le 2$, the space $L_p(\mathbb{R}^d)$ does not admit unconditional Schauder frames $\left\lbrace f_i,f_i'\right\rbrace_{i\in\mathbb{N}}$ where $\left\lbrace f_i\right\rbrace$ is a sequence of translates of finitely many functions and $\left\lbrace f_i'\right\rbrace$ is seminormalized. In fact, the only subspaces of $L_p(\mathbb{R}^d)$ admitting such Banach frames are those isomorphic to $\ell_p$. On the other hand, if $2<p<+\infty$ and $\left\lbrace \lambda_{i}\right\rbrace_{i\in\mathbb{N}}\subseteq \mathbb{R}^d$ is an unbounded sequence, there is a subsequence $\left\lbrace \lambda_{m_i}\right\rbrace_{i\in\mathbb{N}}$, a function $f\in L_p(\mathbb{R}^d)$, and a seminormalized sequence of bounded functionals $\left\lbrace f_i'\right\rbrace_{i\in\mathbb{N}}$ such that $\left\lbrace T_{\lambda_{m_i}}f,f_i'\right\rbrace_{i\in\mathbb{N}}$ is an unconditional Schauder frame for $L_p(\mathbb{R}^d)$.