Functions on the Hamming cube {-1,1}^n can be written as Fourier-Walsh expansions. In this talk, we study an Lp-inequality of Naor relating certain truncations of said Fourier-Walsh expansions, which happen to be conditional expectations, and discrete derivatives. The above result has deep connections with the theory of Lipschitz inclusions between Banach spaces, and it is proven using harmonic analysis tools. We shall investigate Lp-estimates for balanced averages of Fourier truncations in other group algebras, in terms of differential operators acting on them. Our prime example is the free group Fn. Our main inequality relates norms in Lp(LFn), the noncommutative Lp space associated with the group von Neumann algebra of Fn. For our balanced Fourier truncations, we will explore two natural options: conditional expectations and Hilbert transforms. We shall also discuss the right notion of discrete derivative in our group theoretic setting.