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It is known that a system formed by translates of a single function cannot bean unconditional Schauder basis in the space $L^p(\mathbb{R})$ for any $1 \le p< \infty$. To the contrary, there do exist unconditional Schauder frames oftranslates in $L^p(\mathbb{R})$ for every $p>2$. The existence of such a systemfor $1 < p \leq 2$, however, has remained an open problem. In this paper theproblem is solved in the negative: we prove that none of the spaces$L^p(\mathbb{R})$, $1 \le p \le 2$, admits an unconditional Schauder frame oftranslates.

There are no unconditional Schauder frames of translates in $L^p(\mathbb{R})$, $1 \le p \le 2$