February | 2011 | Hydrobates

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Weighted Sobolev spaces

February 22, 2011

In the study of elliptic systems of partial differential equations on compact manifolds it is often helpful to use Sobolev spaces. The Sobolev space $H^s$ is the space of functions which are square integrable together with their derivatives up to order $s$ . In general it is necessary to use distributions and distributional derivatives. Sobolev spaces can be defined in a similar way for sections of vector bundles but I will restrict myself to real-valued functions here. The Sobolev spaces $H^s$ are Hilbert spaces. It is possible to replace $L^2$ by $L^p$ , $1<p<\infty$ in the definitions with the resulting spaces being Banach spaces. On a manifold it is necessary to specify what kind of derivatives are used, how the lengths of these objects are defined and what measure is used for integration. One way of doing this is to use a Riemannian metric defined on the given manifold. Then covariant derivatives can be used, the length of tensors is naturally defined and the metric defines a volume form. In general using different metrics defines norms which are different but equivalent. An alternative approach is based on local coordinates. Introduce a partition of unity each of whose members has a support which is contained in a coordinate chart to write any function $f$ as a sum of functions $f_i$ which can be transported to Euclidean space using the charts. Define the square of the Sobolev norm of $f$ to be equal to the sum of the squares of the corresponding flat space norms of the $f_i$ . Different choices of the partition of unity and the charts lead to norms which are equivalent to each other and to the norms defined by means of Riemannian metrics. In the end the uniqueness of the Sobolev spaces as topological vector spaces (although not as normed spaces) is guaranteed by the compactness of the manifold.

Next some properties of Sobolev spaces on compact manifolds will be listed. If $s_2>s_1$ then the natural embedding from $H^{s_2}$ to $H^{s_1}$ is compact. A linear differential operator of order $k$ whose coefficients belong to a Sobolev space of sufficiently high order defines a bounded linear operator from $H^s$ to $H^{s-k}$ . If the operator is elliptic then the corresponding mapping between Sobolev spaces is Fredholm. This means by definition that its kernel is finite dimensional and that its image is closed and of finite codimension. Elliptic regularity holds: if $P$ is an elliptic operator of order $k$ and $Pu$ belongs to $H^s$ then $u$ belongs to $H^{s+k}$ .

On a non-compact manifold things are more complicated. We have to expect that if there is any reasonable analogue of the theory just sketched it will depend on the asymptotics of the metrics used or the type of coordinate systems chosen. I am familar with one theory of this type which plays a role in general relativity. This is the theory of weighted Sobolev spaces on manifolds with an asymptotically flat metric. Since I was interested in a generalization of this and a bit rusty on the known theory I decided to refresh my memory by rereading the basic references. These are a paper of Choquet-Bruhat and Christodoulou (Acta Math. 146, 129) and one of Bartnik (Commun. Pure Appl. Math. 39, 661). The weighted Sobolev spaces used in this theory differ from ordinary Sobolev spaces by the introduction of positive functions (weights) multiplying the volume form in the definition of the norms. The weights used depend on the choice of a real number $\delta$ in addition to the differentiability index $s$ . The notational conventions used in the two papers just quoted differ and to avoid any ambiguity let me state that in what follows I use Bartnik’s conventions. (They seem to be the ones most frequently encountered in the literature.)

A Riemannian manifold $(M,g)$ is called asymptotically flat if the following conditions hold. There is a compact subset $K$ of $M$ such that the complement of $K$ is diffeomorphic to the complement of a closed ball in a Euclidean space of the same dimension. If the restriction of the metric to the complement of $K$ is transported to $R^n$ using a suitable diffeomorphism of this type then its components in Cartesian coordinates tend to those of the flat Euclidean metric at infinity in a suitable sense. One of the ways of making this into a precise definition is to use weighted Sobolev spaces of functions on flat space. (For the experts I remark that I have restricted here to the case of one end.) The weights are powers of a function $\sigma$ which is everywhere smooth and behaves at infinity like the distance $r$ from a fixed point. These weights are chosen so that (with Bartnik’s conventions) the growth rate of a function in the space $H^s_\delta$ at infinity is no greater than $r^\delta$ and that the growth rate decreases by one for each derivative taken up to order $s$ . These statements are to be understood in an $L^2$ sense and not pointwise. There do, however, exist Sobolev embedding theorems which relate them to pointwise statements. The definitions and a number of the basic properties of these spaces are valid for any complete Riemannian manifold. For instance, the embedding from $H^{s_1}_{\delta_1}$ into $H^{s_2}_{\delta_2}$ is compact if $s_1>s_2$ and $\delta_1<\delta_2$ . The elliptic theory requires more special assumptions. It uses scaling properties of Euclidean space and the fundamental solution for the flat space Laplacian. It can be shown, for instance, that suitable second order elliptic operators with coefficients belonging to weighted Sobolev spaces are Fredholm as operators from $H^{s+2}_\delta$ to $H^s_{\delta-2}$ and all non-integer values of $\delta$ . For integer values of $\delta$ the Fredholm property is lost. This is connected to the following fact. Each Fredholm operator has an index, which is the difference between the dimensions of the kernel and cokernel. The index is invariant under continuous variations of the operator. Changing the coefficient $\delta$ for a fixed elliptic differential operator can be thought of as a continuous variation of the operator between Hilbert spaces. However the index in general changes when passing through an integer value of $\delta$ and so the Fredholm property must be lost there.