Hypercontractivity

A semigroup is hypercontractive when evolution improves integrability through norm bounds $\Vert P_{t}f{\Vert }_{q(t)}\le \Vert f{\Vert }_p$ with $q(t)>p$ for positive times. For a symmetric Markov semigroup satisfying a logarithmic Sobolev inequality with the joint-energy normalization, one may take $q(t)=1+(p-1)e^{4t/c_{\mathrm{LS}}}$.

// Target: analysis.bigb