For Carnot groups, the author proves a \(q\)-logarithmic Sobolev inequality for probability measures as a function of the Carnot-Caratheodory distance. As an application, she uses the Hamilton-Jacobi equation in the setting of Carnot groups to prove the \(p\)-Talagrand inequality and hypercontractivity. More precisely, she shows:
(a) Let \(d \lambda \) denote a measure satisfying the \(q\)-Poincaré inequality, let \(U: [0,\infty) \rightarrow [0,\infty)\) be a twice differentiable and increasing function so that \(d \mu_U= \exp(-U(d))d\lambda\) is a probability measure. Let \(d\) be the Carnot distance. Then, it holds that outside the open unit ball for \(d\), the metric \(d\) satisfies:
a) \( \vert \nabla \vert \) is bounded and there exist finite positive constants \(K,c_0\) such that \(\Delta d\leq K +U'(d)(\vert \nabla d \vert^2 -c_0)\). Then, under a) the author shows:
(i) If \(U'' \leq \beta U'\) (\(\beta >0)\) outside the unit ball for \(d\), then for any \(q \in (1,\infty)\), there exist constants \(C_q, D_q\) such that for every \(f\), \(\int \vert f\vert^q \vert U'(d)\vert d\mu_U \leq C_q \int \vert \nabla f\vert^q d\mu_U + \int \vert f\vert^q d\mu_U \).
(ii) Assume \(d\lambda\) is Lebesgue measure. Let \(1<q\leq 2, p\geq 2 \) so that \(1/p +1/q=1\). If \((G,d,\mu)\) satisfies the \(q\)-logarithmic Sobolev inequality with constant \((q-1)(\frac{q}{K})^{q-1}\) for some \(K>0\), then it also satisfies the \(p\)-Talagrand inequality with the same constant \(K\).