A Beckmann boundary form of Talagrand's conjecture on the discrete cube

Abstract

We introduce the Beckmann boundary of a Boolean function \[ \mathsf{B}(f)=\inf_{\operatorname{div} V=Lf}\mathbb E\|V(x)\|_2. \] Here \[ L=\sum_iD_i,\qquad D_i f(x)=\frac{f(x)-f(x^{\oplus i})}{2}, \] and $\mathrm{div}V(x)={\sum }_i(V_i(x)-V_i(x^{\oplus i}))$. This nonlocal quantity is no larger than the usual two-sided, one-sided, colored, optimized colored, or optimized fractional colored boundaries. Nevertheless, every nonconstant Boolean $f$ satisfies \[ \mathsf{B}(f)\gtrsim \operatorname{Var}(f) \sqrt{\log\!\left(1+\frac{1}{\sum_i\operatorname{Inf}_i(f)^2}\right)}. \] We also prove strong one-sided fractional spectral estimates. If $A\subset \{-1,1{\}}^n$ and \[ h_{A}(x)=\#\{i:x\in A,\ x^{\oplus i}\notin A\}, \] then, for $0<\alpha <1$, \[ \sum_{S\ne\varnothing}|S|^α\widehat{\mathbf 1_{A}}(S)^2 \lesssim_α\mathbb Eω_α(h_{A}), \] where ${\omega }_{\alpha }(m)=\sqrt{m}$ for $\alpha <1/2$, ${\omega }_{1/2}(m)=\sqrt{m}\log (e+m)$, and ${\omega }_{\alpha }(m)=m^{\alpha }$ for $\alpha >1/2$. These profiles are sharp, up to $\alpha$-dependent constants, for majority. We also show that the comparison is genuinely nonreversible: an explicit quotient-cube family makes the optimized fractional, and hence optimized colored, boundary exceed $\mathsf{B}$ by a factor $\gtrsim \sqrt{\log n}$. We further obtain a driftless Bernstein-multiplier inequality.

35 pages, 1 figure