Resolvent kernel estimates near thresholds

Abstract

The paper deals with the spectral structure of the operator $ H=-\nabla\cdot b \nabla $ in $\mathbb R^n$ where $b$ is a stratified matrix-valued function. Using a partial Fourier transform, it is represented as a direct integral of a family of ordinary differential operators $H_p,\, p\in \mathbb{R}^n.$ Every operator $H_p$ has two thresholds and the kernels are studied in their (spectral) neighborhoods, uniformly in compact sets of $p$. As in [3], such estimates lead to a limiting absorption principle for $H$. Furthermore, estimates of the resolvent of $H$ near the bottom of its spectrum ("low energy" estimates) are obtained.