Introduces Beckmann boundary B(f) as inf E||V||_2 over div V = Lf and proves B(f) ≳ Var(f) sqrt(log(1 + 1/sum Inf_i(f)^2)) for nonconstant Boolean f, with sharp one-sided fractional spectral bounds.
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Sharp lower bounds ||S_p(1_A)||_1 ≳ |A|* log(1/|A|*) are established for dyadic square functions S1 and S2 on indicators, using Brownian exit times and the Takagi function.
Dictator functions maximize Φ-stability locally for balanced Boolean functions; computer methods confirm Courtade-Kumar conjecture for ρ≤0.914 and symmetrized Li-Médard for q∈[1.36,2).
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A Beckmann boundary form of Talagrand's conjecture on the discrete cube
Introduces Beckmann boundary B(f) as inf E||V||_2 over div V = Lf and proves B(f) ≳ Var(f) sqrt(log(1 + 1/sum Inf_i(f)^2)) for nonconstant Boolean f, with sharp one-sided fractional spectral bounds.