Pointwise convergence of the product integral for a certain integral transformation associated with a Riemannian metric
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 \[ [(\partial /\partial t)-\lambda^{-1}{\mathcal H}_ x]{\mathbb{H}}(\lambda;t;x,y)=0;\quad \lim_{t\to 0^+}{\mathbb{H}}(\lambda;t;x,y)=\delta y(x)\otimes Idy, \] with \[ {\mathcal H}_ x=(1/2)\Delta^ D_ x-(1/2)\hat r(x),\quad (\Delta^ D=-D^*D), \] where \(D^*\) is the adjoint operator of D with respect to the inner product on \(L^ 2(E)\) and \(\hat r(\)x) is the scalar curvature.
fundamental solution of a parabolic equation
scientific article; zbMATH DE number 3931688