= Logarithmic Sobolev inequality implies hypercontractivity
{title2=$\|P_tf\|_{1+e^{4t/c_{\mathrm{LS}}}}\leq\|f\|_2$}
Differentiate $\log\|u\|_q$ with $u=P_tf$ and a changing exponent. The derivative is $q'\operatorname{Ent}(u^q)/(q^2\int u^q)-\mathcal E(u^{q-1},u)/\int u^q$. Apply the logarithmic Sobolev inequality to $u^{q/2}$ and the <power inequality for symmetric Markov energies>. Choosing $q'=4(q-1)/c_{\mathrm{LS}}$ makes the upper bound zero.
// Target: probability-and-statistics.bigb
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