Ultracontractivity and Nash type inequalities (Q1816722)

A semigroup \(T_t\) acting on the \(L^p\) spaces is said to be ultracontractive if, for every \(t>0\), \(T_t\) sends \(L^1\) into \(L^\infty\): there exists \(m\) from \(\mathbb{R}^*_+\) to itself such that \(|T_t |_{1\to \infty} \leq m(t)\) for all \(t>0\). The author proves that the estimate \(|T_t |_{1\to \infty} \leq m(t)\) is also equivalent to a Nash type inequality for every function \(m\) whose logarithmic derivative has polynomial growth.