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The main aim of this work is to study important local Banach space constantsfor Boolean cube function spaces. Specifically, we focus on$\mathcal{B}_{\mathcal{S}}^N$, the finite-dimensional Banach space of allreal-valued functions defined on the $N$-dimensional Boolean cube $\{-1,+1\}^N$ that have Fourier--Walsh expansions supported on a fixed~family$\mathcal{S}$ of subsets of $\{1, \ldots, N\}$. Our investigation centers onthe projection, Sidon and Gordon--Lewis constants of this function space. Wecombine tools from different areas to derive exact formulas and asymptoticestimates of these parameters for special types of families $\mathcal{S}$depending on the dimension $N$ of the Boolean cube and other complexitycharacteristics of the support set $\mathcal{S}$. Using local Banach spacetheory, we establish the intimate relationship among these three importantconstants.

Asymptotic insights for projection, Gordon-Lewis and Sidon constants in Boolean cube function spaces