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Appendix B Function Spaces

[page 61, §1]

[61.1.1] The set G denotes an interval, a domain in Rd or a measure space G,A,μ [8] depending on the context. [61.1.2] K stands for R or C. γ=γ1,…,γd∈N0d is a multiindex and γ=∑i=1dγi. [61.1.3] For the definition of Hilbert and Banach spaces the reader may consult e.g. [128]. [61.1.4] The following notation is used for various spaces of continuous functions:

C0(G):={f:G→K|f is continuous}(B.1)
Ck⁢G:=f∈C0⁢G|f⁢ is k-times continuously differentiable(B.2)
C0k⁢G:=f∈Ck⁢G|f⁢ vanishes at the boundary ⁢∂⁡G(B.3)
Cbk⁢G:=f∈Ck⁢G|f⁢ is bounded(B.4)
Cck⁢G:=f∈Ck⁢G|f⁢ has compact support(B.5)
Cubk⁢G:=f∈Ck⁢G|f⁢ is bounded and uniformly continuous(B.6)
ACk⁢a,b:=f∈Ck⁢a,b|fk⁢ is absolutely continuous(B.7)

[61.1.5] For compact G the norm on these spaces is

f∞:=supx∈G⁡f⁢x.(B.8)

[61.1.6] The Lebesgue spaces over G,A,μ are defined as

Llocp(G,μ):={f:G→K|fp is integrable on every compact K⊂G}(B.9)
Lp(G,μ):={f:G→K|fp is integrable}(B.10)

with norm

fp:=∫Gf⁢sp⁢d⁢μ⁢s1/p.(B.11)

[page 62, §0]    [62.0.1] For p=∞

L∞⁢G,μ:=f:G→K⁢f⁢ is measurable and ⁢f∞<∞(B.12)

where

f∞:=sup⁡z:z∈fess⁢G(B.13)

and

fess⁢G:=z∈C:μ⁢x∈G:f⁢x-z<ε≠0⁢ for all ⁢ε>0(B.14)

is the essential range of f.

[62.1.1] The Hölder spaces Cα⁢G with 0<α<1 are defined as

Cα(G):={f:G→K|∃c≥0 s.t. |f(x)-f(y)|≤c|x-y|α,∀x,y∈G}(B.15)

with norm

fα:=f∞+cα(B.16)

where cα is the smallest constant c in (B.15). [62.1.2] For α>1 the Hölder space Cα⁢G contains only the constant functions and therefore α is chosen as 0<α<1. [62.1.3] The spaces Ck,α⁢G, k∈N, consist of those functions f∈Ck⁢G whose partial derivatives of order k all belong to Cα⁢G.

[62.2.1] The Sobolev spaces are defined by

Wk,p⁢G=f∈Lp⁢G:f is k-times differentiable in the sense of distributions and Dγ⁢f∈Lp⁢G  for all γ∈N0d with γ≤k (B.17)

where the derivative Dγ=∂1γ1⁡…⁢∂dγd with multiindex γ=γ1,…,γd∈N0d is understood in the sense of distributions. [62.2.2] A distribution f is in Wk,p⁢G if and only if for each γ∈N0d with γ≤k there exists fγ∈Lp⁢G such that

∫Gϕ⁢fγ⁢d⁢x=-1γ⁢∫GDγ⁢ϕ⁢f⁢d⁢x(B.18)

for all test functions ϕ. [62.2.3] In the special case d=1 one has f∈Wk,p⁢G if and only if f∈Ck-1⁢G, fk-1∈AC⁢G, and fj∈Lp⁢G for j=0,1,…,k. [62.2.4] The Sobolev spaces are equipped with the norm

fWk,p⁢G=∑γ≤mDγ⁢fp(B.19)

(see [2]). [62.2.5] A function is called rapidly decreasing if it is infinitely many times differentiable, i.e. f∈C∞⁢Rd and

limx→∞⁡xn⁢Dγ⁢f⁢x=0(B.20)

[page 63, §0]    for all n∈N and γ∈Nd. [63.0.1] The test function space

S⁢Rd:=f∈C∞⁢Rd|f⁢ is rapidly decreasing(B.21)

is called Schwartz space.