Coulhon, Thierry Ultracontractivity and Nash type inequalities. (English) Zbl 0887.58009 J. Funct. Anal. 141, No. 2, 510-539 (1996). 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. Reviewer: J.M.Ayerbe (Sevilla) Cited in 1 ReviewCited in 73 Documents MSC: 58D07 Groups and semigroups of nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. Keywords:semigroup of nonlinear operators; ultracontractivity; Riemannian manifolds; \(L^ p\)-spaces PDF BibTeX XML Cite \textit{T. Coulhon}, J. Funct. Anal. 141, No. 2, 510--539 (1996; Zbl 0887.58009) Full Text: DOI OpenURL