Some classical function spaces

This section briefly introduces the notation for the most frequently used spaces.

Spaces of continuous functions The space of bounded and continuous real- or complex-valued functions on R^d is denoted by BC(R^d). The space of bounded, continuous, real- or complex-valued functions vanishing at infinity is denoted by BC∞(R^d), i.e.

BC_∞(R^d) :=

ϕ∈BC(R^d) : lim

|x|→∞ϕ(x) = 0 . The spaces BC(R^d)and BC∞(R^d) equipped with the supremum norm

kϕk_BC(Rd) := sup

x∈R^d

|ϕ(x)|

are Banach spaces. BC∞(R^d)is a closed subspace ofBC(R^d)arginparA more unique notation would be better We will use

Spaces of differentiable functions To define spaces of differentiable functions the multi-index notation is applied.

Definition 1.2 (Multi-index notation). A multi-index α∈N^d0 is a d-tuple of non-negative integers together with the following set of rules: for α = (α₁, . . . , α_d), β = (β₁, . . . , β_d)∈N^d0 and x∈R^d we define

α≤β :⇔ α1 ≤β1, . . . , αd≤βd, α+β := (α₁+β₁, . . . , α_d+β_d),

|α| := α₁+· · ·+α_d, the order of a multi-index, α! := α₁!. . . α_d!,

x^α := x^α₁¹. . . x^α_d^d.

1.1 Results from functional analysis

the space of real- or complex-valued functions on R^d withk continuous (continuous and bounded) derivatives.

the space of smooth (bounded and smooth), real- or complex-valued functions on R^d.

C_comp^k (R^d) :=

ϕ∈C^k(R^d) | suppϕ⊂R^d is compact ,

the space of k-times continuously differentiable, real- or complex-valued functions with compact support on R^d.

D(R^d) :=C_comp^∞ (R^d) :=

∞

\

k=1

C_comp^k (R^d), the space of test functions.

S(R^d) :=

ϕ∈C^∞(R^d) | ∀m∈N0, β ∈N^d0 : sup

x∈R^d

(1 +|x|^m)|∂^βf(x)|<∞ , the Schwartz space of rapidly decreasing functions. It is obvious that the following inclusion holds

D(R^d)⊂ S(R^d). (1.1)

Spaces of Hölder continuous functions A real- or complex-valued function ϕ defined onR^dis calleduniformly Hölder continuouswithHölder exponent 0< α≤1 if there exists a constant C > 0such that

|ϕ(x)−ϕ(y)| ≤C |x−y|^α (1.2) for allx, y ∈R^d. ByBC^0,α(R^d)we denote the space of all functions that are bounded and uniformly Hölder continuous with exponentα.

BC^0,α(R^d) ⊂ BC(R^d) is called a Hölder space. It is a Banach space equipped with the norm

kϕk_BC^0,α :=kϕk_BC(R^d₎+|ϕ|_α,R^d

where

|ϕ|_α,Rd := sup

x,y∈Rd

x6=y

|ϕ(x)−ϕ(y)|

|x−y|^α

denotes the Hölder semi-norm. |ϕ|_α,Rd is the smallest constant satisfying (1.2) and namedHölder constant of ϕ.

Remark 1.3. In the case α = 1 the function is called Lipschitz continuous. A uniformly Hölder or Lipschitz continuous function is uniformly continuous.

We also need spaces of bounded continuously differentiable functions that have bounded and uniformly Hölder continuous first derivatives. This space is denoted byBC^1,α(R^d). Among the many equivalent norms on BC^1,α(R^d) we choose

kϕk_BC^1,α :=kϕk_BC(Rd)+ sup

x∈R^d

|∇ϕ(x)|+ sup

x,y∈Rd

x6=y

|∇ϕ(x)− ∇ϕ(y)|

|x−y|^α .

The space BC^1,α(R^d)⊂BC¹(R^d) equipped with the norm k · k_BC^1,α is a Banach space.

Spaces of Lebesgue integrable functions For1 ≤p <∞ we denote by L^p(R^d) the linear space of equivalence classes of real- or complex-valued Lebesgue measur-able functionsϕonR^dsuch that|ϕ|^p is integrable overR^d. As it is the usual custom two functions are identified if they agree except for a set of measure zero which is written shortly as a.e. for almost everywhere, i.e. if for two measurable functions defined a.e.

f :R^d\N_f →K, g :R^d\N_g →K,

the setsN_f, N_g and{x∈R^d\(N_f∪N_g) :f(x)6=g(x)}are of measure zero. One can show that all functions in one equivalence class are integrable and have the same integral. On the set of equivalence classes

kϕk_L^p_(R^d₎ :=

Z

R^d

|ϕ(x)|^p dx 1/p

. (1.3)

defines a norm which is called the L^p-norm.

Theorem 1.4 (Riesz-Fischer theorem for 1≤ p < ∞). For 1 ≤ p < ∞ the spaces L^p(R^d) equipped with k · k_L^p_(R^d₎ are Banach spaces. For every convergent sequence there exists a subsequence that converges pointwise almost everywhere.

The linear space of equivalence classes of measurable functions that areessentially bounded is denoted by L^∞(R^d), i.e. a measurable function ϕ : R^d → K belongs to

1.1 Results from functional analysis

L^∞(R^d) if

kϕk_L^∞_(Rd) := ess. sup

x∈R^d

|ϕ(x)|:= inf

C ∈R : |ϕ(x)| ≤C a.e. <∞. (1.4) One can show that kϕk_L^∞_(R^d₎ defines a norm onL^∞(R^d).

Theorem 1.5 (Riesz-Fischer theorem for p = ∞). The spaces L^∞(R^d) equipped with k · k_L^∞_(R^d₎ is a Banach spaces. For every convergent sequence there exists a subsequence that converges pointwise almost everywhere.

Remark 1.6. Following the usual custom we talk ofL^p-functions instead of equiva-lence classes of functions. This is a reasonable approach as long as one does not try to use pointwise properties of L^p-functions. One exception is the case that f ∈ L^p equals to a continuous function a.e.. As one can show that there can only be one continuous function in one equivalence class, one chooses this functions as repre-sentative.

To prove mapping properties of convolution operators onL^p-spaces inSection 1.5 we need the Hölder inequality. It is used also to prove the Minkowski inequality, i.e. to show that (1.3) actuallyis a norm.

In dealing withL^p-spaces it is useful to define conjugate pairs of exponents, i.e. for p∈[1,∞]let p⁰ denote theconjugate exponent such that

1 p+ 1

p⁰ = 1 where one uses the convention _∞¹ = 0.

Theorem 1.7 (Hölder inequality). Let 1≤p≤ ∞. For f ∈L^p(R^d), g ∈L^p0(R^d) it holds that f g ∈L¹(R^d) and

kf gk_L¹ ≤ kfk_L^p kgk_Lp0 (1.5) or written as integral inequality

Z

R^d

|f(x)g(x)|dx ≤ Z

R^d

|f(x)|^p dx

1/p Z

R^d

|g(x)|^p0 dx 1/p⁰

. Proof. For a proof see [58, Satz 1.39].

An easy consequence is the

Corollary 1.8 (Generalised Hölder inequality). Let 1 ≤ p, q, r ≤ ∞ with ¹_r =

1

p + ¹_q. For f ∈L^p(R^d), g∈L^q(R^d) it holds that f g∈L^r(R^d) and

kf gk_L^r ≤ kfk_L^p kgk_L^q (1.6)

or written as integral inequality

Thus we can apply the ordinary Hölder inequality (Theorem 1.7) and compute Z Taking therth root on both sides finishes the proof.

In the case that the Lebesgue spaces are defined over some bounded domain Ω⊂R^d it is a consequence of the Hölder inequality that for1< p < q≤ ∞it holds a very elementary example of a so called interpolation theorem, sometimes called Lyapunov inequality cf. [60, Lemma II.4.1].

Lemma1.9. Let 1≤p≤q≤ ∞ and0≤θ≤1. Define sthrough ¹_s = (1−θ)¹_p+θ¹_q, Taking thes-th root on both sides finishes the proof.

1.1 Results from functional analysis

The following example illustrates the preceding Lemma. We use it later on to show the mapping properties of the boundary operators.

Example 1.10. Consider the function on R² given through

`(y) :=

(|y|^α−2, |y| ≤1, 0, |y|>1,

for some α∈(0,1]. Changing to polar coordinates one computes that Z

R²

|`(y)|^p dy = Z

B1(0)

|`(y)|^p dy= 2π Z 1

0

r^(α−2)pr dr <∞ for all p∈[1,_2−α² ), i.e. `∈L¹(R²)∩L^p(R²) for 1≤p < _2−α² <2.

For later use it is important to know dense subspaces of Lebesgue spaces. One can show

Lemma 1.11. For 1≤p < ∞the spaceD(R^d)is a dense subspace of L^p(R^d), i.e. for each ϕ ∈ L^p(R^d) there exists a sequence of functions (ϕ_n)n∈N in D(R^d) such that kϕ−ϕk_L^p_(R^d₎ →0 for n → ∞.

As an immediate consequence of the inclusion (1.1) we get:

Lemma 1.12. For 1≤p < ∞ the space S(R^d) is a dense subspace of L^p(R^d).

Among the Lebesgue space the case p = 2 deserves some extra attention as it is the only L^p-space that is a Hilbert space. The scalar product is given in the usual way by

(ϕ, ψ)_L²_(R^d₎ :=

Z

R^d

ϕ(x)ψ(x)dx,

for ϕ, ψ ∈ L²(R^d) and we see that the Cauchy-Schwarz inequality is just a special case of the Hölder inequality, i.e.

|(ϕ, ψ)_L²| ≤ kϕψk_L¹ ≤ kϕk_L²kψk_L².

Radial functions For later use in connection with symmetry properties of Fourier transformation it is useful to introduce the concept of radial functions inL^p-spaces.

A function f :R^d→Kis said to be radial if it can be written in the form f(x) =f_rad(|x|), x∈R^d,

where f_rad :R^≥0 →C. It is clear that a function is radial if and only if f(x) = f(Qx), for all x∈R^d and Q∈O(R^d),

whereO(R^d)denotes the group of orthogonal transformations onR^d. Iff ∈L^p(R²)is a radial function it follows from Fubini’s theorem together with a change of variables x=r(cosθ,sinθ)that

kfk^p_Lp = Z

R²

|f(x)|^p dx=

∞

Z

0 2π

Z

0

|f_rad(r)|^p dθ rdr = 2π

∞

Z

0

|f_rad(r)|^p rdr <∞,

i.e. the radial part of f is an element ofL^p(R^≥0, rdr), the weightedL^p-space on the positive half-line.

Im Dokument Integral Equation Methods for Rough Surface Scattering Problems in three Dimensions (Seite 32-38)