There is a certain duality between estimates a) on the Hamming involving functions and their discrete gradients, and b) for dyadic martingales and corresponding square functions. Roughly speaking one can take a valid estimate for the dyadic square function, ``dualize''  it by a certain double Legendre transform, and one can obtain its corresponding dual estimate on the Hamming cube, and vice versa. I will illustrate on concrete examples how this duality works, and we will see that by dualizing certain sharp martingale estimates of Burkholder, Davis, Gundy we can improve several results on the  Hamming cube for free (joint work with F. Nazarov and A. Volberg).