Pointwise convergence of the product integral for a certain integral transformation associated with a Riemannian metric
type: Item
instance of: scholarly article
title: Pointwise convergence of the product integral for a certain integral transformation associated with a Riemannian metric
zbMATH Open document ID: 0582.44002
author: Yoshiaki Maeda
published in: Kodai Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: Let M be an m-dimensional connected, simply connected \(C^{\infty}\) manifold with metric g. Denoting by \(\nabla\) and \(R^ h_{kji}\) the Riemann connection of M determined by g and the curvature tensor of \(\nabla\), respectively, we suppose that for each point x of M, \(| \nabla^{\alpha}R^ h_{kji}| \leq k_ 1\), where \(\alpha =(\alpha_ 1,...,\alpha_ m)\), \(0\leq | \alpha | <3\) and \(| |_ x\) denotes the norm at x defined by g. Let E be a vector bundle over M with a linear connection D, and \(<, >_ x\) be an inner product at each fibre \(E_ x\) preserved by D. With D we can define the parallel translation P(x,y) from y to x along a minimizing geodesic. We suppose for the curvature form \(\Omega\) of D that \(| D^{\alpha}\Omega (x)|_ x<k_ 2\) for \(0\leq | \alpha | <3\). Let \(C_ 0^{\infty}(E)\) be the set of smooth sections of E, and define \(L^ 2\)- norm of \(\xi \in C_ 0^{\infty}(E)\) by \[ \| \xi \|_{L^ 2(E)}=[\int_{M}<\xi (x),\xi (x)>_ xd\mu_ g(x)]^{1/2}, \] where \(d\mu_ g(x)\) is the canonical measure defined by g. Denoting by \(L^ 2(E)\) the Hilbert space of sections \(\xi\) of E such that \(\| \xi \|_{L^ 2(E)}<+\infty,\) one defines for \(\xi\) an integral transformation in \(L^ 2(E)\) by \[ H(\lambda;t,s)\xi (x)=(2\pi \lambda^{-1})^{-m/2}\int_{M}\rho (t,s;x,y)[\exp (-\lambda S(t,s;x,y))]\times P(x,y)\xi (y)d\mu_ g(y), \] where \[ S(t,s;x,y)=d^ 2(x,y)/(2(t-s)),\quad (0\leq s<t;\quad x,y\in M), \] and \[ S(t,s;x,y)=\det [-\partial_ x\partial_ yS(t,s;x,y)\mu_ g(x)\mu_ g(y)]^{1/2}. \] By taking the N equal-subdivision \(\sigma_ N\) of the interval [0,t], we denote the product integral \(H(\lambda;t,t_{N- 1})H(\lambda;t_{N-1},t_{N-2})...H(\lambda;t_ 1,0)\) by \(H(\lambda;\sigma_ N| t)\). Then it can be proven that for a fixed \(T>0\), the limit kernel function \({\mathbb{H}}(\lambda;t;x,y)=\lim H(\lambda;\sigma_ N| t;x,y)\) converges uniformly on \(M\times M\) in the norm defined by \(E\otimes E^*\), giving a fundamental solution of a parabolic equation \[ […
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 35K10
zbMATH DE Number: 3931688
zbMATH Keywords: Riemann connection
zbMATH Keywords: curvature tensor
zbMATH Keywords: vector bundle
zbMATH Keywords: linear connection
zbMATH Keywords: Hilbert space of sections
zbMATH Keywords: integral transformation
zbMATH Keywords: product integral
zbMATH Keywords: limit kernel function
zbMATH Keywords: fundamental solution of a parabolic equation
zbMATH Keywords: scalar curvature
reviewed by: Tanjiro Okubo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037047
OpenAlex ID: W2013248466
Wikidata QID: Q115224629
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DOI: 10.2996/KMJ/1138037047
description: scientific article; zbMATH DE number 3931688