PapersThe multislice is a generalization of the slice. Whereas the slice consists of all vectors in $\{0,1\}^n$ of fixed Hamming weight, the multislice consists of all vectors in $[\ell]^n$ of fixed histogram.
The log-Sobolev inequality is a fundamental inequality in Boolean Function Analysis, equivalent to hypercontractivity. Lee and Yau determined (up to a constant factor) the optimal constant in the log-Sobolev inequality for the slice. We give a clear exposition of their argument, and extend it to the multislice.
As applications, we derive versions of the Kahn–Kalai–Linial, Friedgut’s junta, Kruskal–Katona, and Nisan–Szegedy theorems.
Our log-Sobolev inequality is tight only for constant $\ell$. Justin Salez has since proved a tight log-Sobolev inequality for all multislices.
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