Global heat kernel bounds via desingularizing weights
Let \(A\geq 0\) be a self-adjoint operator in \(L^2(M,d\mu)\) such that \({\mathcal D}(A^{1/2})\subset L^{2j}, j > 1\). The authors assume that there exists a family of weights \(\phi_t\) on \(M\) such that \(\phi_s, 1/\phi_s \in L^2(M,d\mu)\) for all \(s>0\), and that for all \(0<t\leq s\) \[ \| \phi_s e^{-tA}\phi_s^{-1} f\| _1 \leq c\| f\| _1,\;\forall f\in\phi_s L^\infty_{\text{comp}}(M,d\mu). \] Under the additional assumption that \(\inf_{s>0,x\in M}| \phi_s(x)| >0\), the following upper bound is proved: \[ | e^{-tA}(x,y)| \leq Ct^{-j'}| \phi_t(x)\phi_t(y)| , \;j'={j\over{j-1}}. \] The proof is based upon an abstract version of the approach of \textit{J. Nash} [Am. J. Math. 80, 931--954 (1958; Zbl 0096.06902)]. In the framework of symmetric Markov semigroups, the authors also provide lower bounds for \(e^{-tA}(x,y)\). An application to the Schrödinger operators \(-\Delta-\beta V\) with critical potential \(V={{(d-2)^2}\over 4}| x| ^{-2}, \beta\in[0,1]\) is included. The erratum corrects a misleading reference in the proof for the limit case \(\beta=1\) as approx. one page of the original manuscript was not included in the printed article.
scientific article; zbMATH DE number 2105598
scientific article