Decay transference and Fredholmness of differential operators in weighted Sobolev spaces

Abstract

We show that, for some family of weights $\omega ,$ there are corresponding weighted Sobolev spaces $W_{\omega }^{m,p}$ on $ \mathbb {R}^{N}$ such that whenever $P(x,\partial)$ is a differential operator with $L^{\infty }$ coefficients and $P(x,\partial):W^{m,p}\rightarrow L^{p}$ is Fredholm for some $p\in (1,\infty),$ then $P(x,\partial):W_{\omega }^{m,p}\rightarrow L_{\omega }^{p}$ ($=W_{\omega }^{0,p}$) remains Fredholm with the same index. We also show that many spectral properties of $P(x,\partial)$ are closely related, or even the same, in the non-weighted and the weighted settings. The weights $\omega $ arise naturally from a feature of independent interest of the Fredholm differential operators in classical Sobolev spaces (``full'' decay transference), proved in the preparatory Section 2. A main virtue of the spaces $W_{\omega }^{m,p}$ is that they are well suited to handle nonlinearities that may be ill-defined or ill-behaved in non-weighted spaces. Together with the invariance results of this paper, this has proved to be instrumental in resolving various bifurcation issues in nonlinear elliptic PDEs.