We develop a new class of log-Sobolev inequalities (LSIs) that provide a non-linear comparison between the entropy and
the Dirichlet form. For the hypercube, these LSIs imply a new (and in essence optimal) version of the hypercontractivity
for functions of small support. As a consequence, we derive a sharp form of the uncertainty principle for the hypercube:
a function whose energy is concentrated on a set of small size, and whose Fourier energy is concentrated on a small
Hamming ball must be zero. The tradeoff we derive is asymptotically optimal. As an application, we show how uncertainty
principle implies a new estimate of the metric properties of linear maps $mathbb{F}_2^ktomathbb{F}_2^n$.