Heat kernel estimates for an operator with unbounded diffusion coefficients in \(\mathbb R\) and \(\mathbb R^{2}\)
type: Item
instance of: scholarly article
title: Heat kernel estimates for an operator with unbounded diffusion coefficients in \(\mathbb R\) and \(\mathbb R^{2}\)
zbMATH Open document ID: 1272.35111
published in: Semigroup Forum
publication date: 2013-03-21T00:00:00Z
review text: The author considers the operator \(L = (1 + |x|^2)^{\alpha/2} \Delta\) for \(\alpha > 0\) in the whole space \(\mathbb{R}^N\) for \(N=1\) and \(N=2\). It is known that in these cases \(L\) generates a semigroup \((T(t))_{t \geq 0}\) in \(C_b(\mathbb{R}^N)\) which can represented by an integral kernel \[ T(t)f(x) = \int_{\mathbb{R}^N} p(x,y,t) f(y) \, dy, \qquad f \in C_b(\mathbb{R}^N). \] The author establishes pointwise upper bounds for \(p\), which were already obtained in [\textit{G. Metafune} and the author, Discrete Contin. Dyn. Syst. 32, No. 6, 2285--2299 (2012; Zbl 1252.47038)] for the case \(N \geq 3\). To exemplify the obtained results, we cite the obtained bounds for \(p\) in the case \(N=2\). In this case one has \[ p(x,y,t) \leq \frac{C}{t} (1 + |y|^2)^{-\alpha/2} \qquad \text{for } 0 < t \leq 1, x,y \in \mathbb{R}^2. \]
reviewed by: Stephan Fackler
Mathematics Subject Classification ID: 35K08
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 6147891
zbMATH Keywords: Nash inequality
zbMATH Keywords: pointwise upper bounds
author: Chiara Spina
MaRDI profile type: Publication
full work available at URL: s00233-012-9420-4
OpenAlex ID: W1987184252
cites work: Q3676658
cites work: Maximal regularity in \(L^p(\mathbb R^N)\) for a class of elliptic operators with unbounded coefficients.
cites work: Q4041031
cites work: Heat kernel estimates for some elliptic operators with unbounded diffusion coefficients
cites work: Elliptic operators with unbounded diffusion coefficients in \(L^p\) spaces
cites work: Feller semigroups on \(\mathbb{R}^N\)
cites work: Q4674789
DOI: 10.1007/S00233-012-9420-4
description: scientific article; zbMATH DE number 6147891