Lecture Notes on Function Spaces
Lecture Notes on Function Spaces space invaders Karoline Götze TU Darmstadt, WS 2010/2011 Contents 1 Basic Notions in Interpolation Theory 5 2 The K-Method 15 3 The Trace Method 19 3.1 Weighted Lp spaces . 19 3.2 The spaces V (p, θ, X0,X1) ........................... 20 3.3 Real interpolation by the trace method and equivalence . 21 4 The Reiteration Theorem 26 5 Complex Interpolation 31 5.1 X-valued holomorphic functions . 32 5.2 The spaces [X, Y ]θ and basic properties . 33 5.3 The complex interpolation functor . 36 5.4 The space [X, Y ]θ is of class Jθ and of class Kθ................ 37 6 Examples 41 6.1 Complex interpolation of Lp -spaces . 41 6.2 Real interpolation of Lp-spaces . 43 6.2.1 Lorentz spaces . 43 6.2.2 Lorentz spaces and the K-functional . 45 6.2.3 The Marcinkiewicz Theorem . 48 6.3 Hölder spaces . 49 6.4 Slobodeckii spaces . 51 6.5 Functions on domains . 54 7 Function Spaces from Fourier Analysis 56 8 A Trace Theorem 68 9 More Spaces 76 9.1 Quasi-norms . 76 9.2 Semi-norms and Homogeneous Spaces . 76 9.3 Orlicz Spaces LA ................................ 78 9.4 Hardy Spaces . 80 9.5 The Space BMO ................................ 81 2 Motivation Examples of function spaces: n 1 α p k,p s,p s C(R ; R),C ([−1, 1]),C (Ω; C),L (X, ν),W (Ω),W ,Bp,q, n where Ω domain in R , α ∈ R+, 1 ≤ p, q ≤ ∞, (X, ν) measure space, k ∈ N, s ∈ R+.
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