Global heat kernel bounds via desingularizing weights
type: Item
instance of: scholarly article
title: Global heat kernel bounds via desingularizing weights
zbMATH Open document ID: 1057.47043
author: Yanyan Li
published in: Journal of Functional Analysis
publication date: 2004-10-01T00:00:00Z
full work available at URL: http://cds.cern.ch/record/630833
review text: 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.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 2105598
zbMATH Keywords: heat kernel bounds
zbMATH Keywords: symmetric Markov semigroups
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: critical potential
zbMATH Keywords: Mazýa--Hardy inequality
MaRDI profile type: Publication
OpenAlex ID: W2032154264
cites work: The Heat Equation with a Singular Potential
cites work: Semigroups on nonlinear contractions on convex sets
cites work: Q4378149
cites work: Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier
cites work: Q4041031
cites work: \(L^ p\) norms of non-critical Schrödinger semigroups
cites work: Q3874765
cites work: Schrödinger operators with l l/2w(Rl) -potentials
cites work: Q4896565
cites work: Q4693168
cites work: Heat kernel bounds and desingularizing weights.
cites work: Continuity of Solutions of Parabolic and Elliptic Equations
cites work: Q4283334
cites work: Q4352610
cites work: Q4225231
cites work: Q4002250
cites work: Q5531981
cites work: Global bounds of Schrödinger heat kernels with negative potentials
DOI: 10.1016/J.JFA.2003.12.008
DOI: 10.1016/J.JFA.2004.10.010
Recommended article: Compactness of Schrödinger semigroups with unbounded below potentials
Recommended article: Q4349048
Recommended article: Q3842444
Recommended article: The maximum Markovian self-adjoint extensions of generalized Schrödinger operators
Recommended article: ASYMPTOTICALLY EXACT ESTIMATES FOR METASTABLE MARKOV SEMIGROUPS
Recommended article: Improved time-decay for a class of scaling critical electromagnetic Schrödinger flows
Recommended article: Q3482184
Recommended article: \(L^p\)-independence of spectral bounds of Schrödinger type semigroups
Recommended article: Convergence of Dirichlet forms and associated Schrödinger operators
Recommended article: Q3839242
description: scientific article; zbMATH DE number 2105598