## 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

### MSC:

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