# Several open problems on the Hamming cube II.

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The Hamming cube of dimension n consists of vectors of length n with coordinates +1 or -1. Real-valued functions on the Hamming cube equipped with uniform counting measure can be expressed as Fourier--Walsh series, i.e., multivariate polynomials of n variables +1 or -1. The degree of the function is called the corresponding degree of its multivariate polynomial representation. Pick a function whose Lp norm is 1. How large the Lp norm of the discrete (graph) Laplacian of the function can be if its degree is at most d, i.e., it lives on ``low frequencies''? Or how small it can be if the function lives on high frequencies, i.e., say all low degree terms (lower than d) are zero? I will sketch some proofs based on joint works (some in progress) with Alexandros Eskenazis.