A Lower Bound on the Trace Norm of Boolean Matrices and its Applications

We present a simple method based on a variant of Hölder’s inequality to lower-bound the trace norm of Boolean matrices. As the main result, we obtain an exponential separation between the randomized decision tree depth and the spectral norm (i.e. the Fourier L₁ -norm) of a Boolean function. This answers an open question of Cheung, Hatami, Hosseini, and Shirley (CCC 2023). As immediate consequences, we obtain the following results.