Sharp constant κ in the L log(p/2)L inequality on the Hamming cube

Determine the sharp (optimal) constant κ(p) in the inequality ||g||_{L log(p/2)L} ≤ κ(p) E[M g] for all 1 < p < ∞ and all nonnegative functions g: {−1,1}^n → ℝ satisfying P{g(x) = 0} ≥ δ, where M g(x) := (∑_{i=1}^n ((D_i g(x))_+)^2)^{1/2} and D_i g(x) := (g(x) − g(x^{(i)}))/2 with x^{(i)} denoting x with the i-th coordinate flipped.

Background

The paper leverages discrete logarithmic Sobolev inequalities to obtain measure concentration results for vector-valued functions on the Hamming cube. A key ingredient is an inequality that bounds the Orlicz norm L log(p/2)L of a nonnegative function by a Talagrand-type gradient functional M g derived from discrete derivatives D_i.

In Proposition 3.2, the authors invoke an inequality of this form for general 1 < p < ∞. While they proceed using only the p = 2 case (leading to Proposition 3.5), they explicitly note that the sharp value of the constant κ in the general inequality is not known. Identifying the optimal κ(p) would refine the constants in concentration bounds and potentially improve related estimates.