## 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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