Wsections

Wsections are a concept in differential geometry and analysis used to denote a class of sections of a vector bundle that are controlled by an external weight function W. Formally, let E → M be a smooth vector bundle over a Riemannian manifold M, and let W: M → (0, ∞) be a smooth weight. A smooth section s of E is called a Wsection if s lies in the weighted L^2 space L^2_W(M; E), meaning that ∫_M W(x) |s(x)|^2 dvol_M < ∞. More generally, one defines weighted Sobolev spaces H^k_W(M; E), and a Wsection may be taken to have k derivatives in L^2_W.

The choice of weight W encodes spatial decay or growth. If W grows at infinity, Wsections require faster decay of s; if W decreases, the condition is weaker. On noncompact manifolds, Wsections provide a convenient framework for studying elliptic operators and variational problems where noncompactness would otherwise obstruct compactness.

Typical examples include choosing W(x) = (1 + d(x, x0)^2)^α on M = R^n with the trivial bundle, in which case a Wsection is a vector-valued function s with ∫ (1 + |x|^2)^α |s(x)|^2 dx < ∞. Such spaces are used to formulate weighted elliptic estimates and to define indices in settings with noncompact ends.

Applications appear in geometric analysis, gauge theory, and mathematical physics, where decay conditions at infinity are essential. See also weighted Sobolev spaces, elliptic operators, and noncompact manifold analysis.