Nash inequalities for Markov processes in dimension one (Q1601228)

The so-called Nash inequality for a symmetric form \(D\) with a domain \(Q\) in \(L^2(\mu)\) has the form \(\|f\|_2^{2+4/\nu}\leq c D(f,f) \|f\|_1^{4/\nu}\), for \(\nu>2\) and \(f\in\{f\in Q:\mu(f\neq 0)<\infty\}\). The aim of the paper is to give some sufficient and necessary conditions of Nash inequalities for Markov chains and diffusions on the line. Certain results for the general case of \(D\) were obtained by \textit{M. Chen} [Acta Math. Sin., Engl. Ser. 15, 353-370 (1999; Zbl 0938.60067)]. The main tools applied in the present paper are Hardy-type inequalities on the line and the equivalence between the Nash inequality and a certain Sobolev type inequality. Sufficient conditions for general Markov chains are also obtained.