Sobolev Spaces (Crash Course)

Abstract

How can we fit the analytic concept of “differential operators” and “pseudo-differential operators” into the function analytic framework of “Hilbert space theory” (see Part I)? There is first of all the L²- concept of a Lebesgue measurable square integrable function which can be transferred naturally to sections in a Hermitian vector bundle E on a Riemannian manifold X: A section u: X → E (not necessarily continuous) represents an element of L² (E) if γ_X <u, u> < ∞ Here <...,...> is a Hermitian metric for the vector bundle E, i.e., <u, u> is a complex valued function on X which is integrated with respect to the volume element defined by the Riemannian structure of X (see above Ch. 2, Section G). In this way, L² (E) becomes a Hibert space with the usual identifications.