Upper bounds for symmetric Markov transition functions. (English) Zbl 0634.60066

Let (E,\({\mathcal B},m)\) be a locally compact separable metric space with a locally finite measure m, \(\{\bar P{}_ t\), \(t\geq 0\}\) be a strongly continuous semigroup of self-adjoint contractions on \(L^ 2(m)\) and (\({\mathcal E},{\mathcal D}({\mathcal E}))\) be an associated Dirichlet form. Nash- type inequalities are proved. Denote \(\| f\|_ p\) the \(L^ p(m)\)- norm of f and \(\| \bar P_ t\|_{p\to q}=\sup \{\| \bar P_ tf\|_ q:\| f\|_ p=1\), f is Borel with compact support\(\}\). Let \(\nu\in (0,\infty)\) and \(\delta\in [0,\infty)\) be fixed. If \[ (*)\quad \| f\|_ 2^{2+4\nu}\leq A[{\mathcal E}(f,f)+\delta \| f\|^ 2_ 2]\| f\|_ 1^{4/\nu},\quad f\in L^ 2(m), \] for some \(A\in (0,\infty)\), then \[ \| \bar P_ t\|_{1\to \infty}\leq Be^{\delta t}/t^{\nu /2},\quad t>0, \] for some \(B\in (0,\infty)\) depending only on \(\nu\) and A and vice versa. If \(\nu\in (2,\infty)\), these inequalities are equivalent to the Sobolev-type inequality \[ \| f\|^ 2_ p\leq A'({\mathcal E}(f,f)+\delta \| f\|^ 2_ 2);\quad p=2\nu /\nu -2. \] Cases when \(\delta =0\) or \(\nu =2\) as well as discrete-time analogues are also considered. Under assumption (*) the semigroup \(\{\bar P{}_ t\), \(t\geq 0\}\) possesses a kernel p(t,x,y) with respect to m. E. B. Davies’ method is extended for obtaining diagonal estimates of p(t,x,y).
Reviewer: B.Grigelionis


60J35 Transition functions, generators and resolvents
60J99 Markov processes
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