## Differential and Integral Equations

- Differential Integral Equations
- Volume 21, Number 11-12 (2008), 1001-1018.

### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 21, Number 11-12 (2008), 1001-1018.

**Dates**

First available in Project Euclid: 14 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1355502291

**Mathematical Reviews number (MathSciNet)**

MR2482494

**Zentralblatt MATH identifier**

1224.35137

**Subjects**

Primary: 35P05: General topics in linear spectral theory

Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

#### Citation

Rabier, Patrick J. Decay transference and Fredholmness of differential operators in weighted Sobolev spaces. Differential Integral Equations 21 (2008), no. 11-12, 1001--1018. https://projecteuclid.org/euclid.die/1355502291