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Ultracontractivity and Nash type inequalities. (English) Zbl 0887.58009

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.

MSC:

58D07 Groups and semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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