By Emmanuel Hebey (auth.)
Several books care for Sobolev areas on open subsets of R (n), yet none but with Sobolev areas on Riemannian manifolds, even though the speculation of Sobolev areas on Riemannian manifolds already is going again approximately two decades. The publication of Emmanuel Hebey will fill this hole, and turn into an important analyzing for all utilizing Sobolev areas on Riemannian manifolds.
Hebey's presentation is especially particular, and comprises the newest advancements due quite often to the writer himself and to Hebey-Vaugon. He makes quite a few issues extra distinctive, and discusses the hypotheses to check whether or not they should be weakened, and in addition offers new results.
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Extra resources for Sobolev Spaces on Riemannian Manifolds
Sample text
Die Auswahl an Begriffen und Sätzen ist jedoch so gewählt, dass auch ohne weiteres Literaturstudium die nachfolgenden Kapitel verstanden werden können. Wir betrachten lineare Abbildungen T : X → Y . Hierbei bezeichnen (X, · X ) und (Y, · Y ) normierte Räume über demselben Körper K (= R ∨ C). T heißt beschränkt := ∃ C ≥ 0 ∀ u ∈ X : Tu Y ≤C u X . 14) Mit diesem Begriff kann die Stetigkeit linearer Operatoren charakterisiert werden. 24 Für lineare Operatoren T : X → Y sind die folgenden drei Bedingungen äquivalent: (i) T ist stetig auf ganz X.
47 X := Y := l2 und T (x1 , x2 , x3 , . ) := (0, x1 , x2 , . ) die Rechtsverschiebung. Da T injektiv ist, folgt dim ker(T ) = 0. Andererseits ist codim R(T ) = 1 und somit ind(T ) = −1. Man überlegt sich leicht, dass ind(T n ) = −n (n ∈ N) gilt. Wir betrachten ein Beispiel eines linearen stetigen Operators, der kein FredholmOperator ist. 52 Es seien X = Y := C 0 [−1, 1] und beide Räume mit der Max-Norm ausgestattet. Der Operator T sei definiert durch s T :X→Y , (T x)(s) := x(t) dt −1 (s ∈ [−1, 1]) .
Iii) T ist stetig in x0 ∈ X. Damit haben wir ein bemerkenswertes Ergebnis: Ein linearer Operator ist entweder in jedem Punkt stetig oder in jedem Punkt unstetig. Wir setzen L(X, Y ) := { T : X → Y | T ist linear und beschränkt } und nennen L(X, Y ) den Vektorraum der linearen stetigen Operatoren. Es lässt sich nämlich leicht zeigen, dass L(X, Y ) mit den üblichen Operationen der Addition „+“ und der skalaren Multiplikation „·“ ein Vektorraum ist. Wir führen nun in L(X, Y ) eine geeignete Norm ein.