There are no unconditional Schauder frames of translates in L^p(R), 1 ⩽ p ⩽ 2

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