Heat kernel estimates for some elliptic operators with unbounded diffusion coefficients (Q423875)

By using the equivalence between Nash inequalities and ultracontractivity for symmetric Markov semigroups, the authors obtain upper bounds for the integral kernel \(p_{\mu}\) in \(L_{\mu}^2\) space and the integral kernel \(p\) with respect to the Lebesgue measure of the semigroup \((T(t))_{t\geq 0}\) generated by the operator \(L=m(x)(1+|x|^{\alpha})\Delta\) in the whole space \(\mathbb{R}^N\), \(N\geq 3\). Here, \(m\) is a bounded locally Hölder continuous function with \(\inf\,m>0\), \(\alpha\) is a positive real number, the measure \(d\mu\) is \(d\mu(x)=(m(x)(1+|x|^{\alpha}))^{-1}\,dx\), and \(p_{\mu}(x,y,t)=m(y)(1+|y|^{\alpha})p(x,y,t)\).