Improved log-Sobolev inequalities, hypercontractivity and uncertainty principle on the hypercube

Log-Sobolev inequalities (LSIs) upper-bound entropy via a multiple of the Dirichlet form (i.e. norm of a gradient). In this paper we prove a family of entropy-energy inequalities for the binary hypercube which provide a non-linear comparison between the entropy and the Dirichlet form and improve on the usual LSIs for functions with small support. These non-linear LSIs, in turn, imply a new version of the hypercontractivity for such functions. As another 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 between the sizes that we derive is asymptotically optimal. This new uncertainty principle implies a new estimate on the size of Fourier coefficients of sparse Boolean functions. We observe that an analogous (asymptotically optimal) uncertainty principle in the Euclidean space follows from the sharp form of Young's inequality due to Beckner. This hints that non-linear LSIs augment Young's inequality (which itself is sharp for finite groups).