Chapter 33

Fourier Analysis In Rn

The purpose of this chapter is to present some of the most important theorems on Fourieranalysis in Rn. These theorems are the Marcinkiewicz interpolation theorem, the CalderonZygmund decomposition, and Mihlin’s theorem. They are all fundamental results whoseproofs depend on the methods of real analysis.

33.1 The Marcinkiewicz Interpolation TheoremLet (Ω,µ,S ) be a measure space.

Definition 33.1.1 Lp (Ω)+L1 (Ω) will denote the space of measurable functions, f , suchthat f is the sum of a function in Lp (Ω) and L1 (Ω). Also, if T : Lp (Ω)+L1 (Ω)→ spaceof measurable functions, T is subadditive if

|T ( f +g)(x)| ≤ |T f (x)|+ |T g(x)|.

T is of type (p, p) if there exists a constant independent of f ∈ Lp (Ω) such that

||T f ||p ≤ A∥ f∥p, f ∈ Lp (Ω).

T is weak type (p, p) if there exists a constant A independent of f such that

µ ([x : |T f (x)|> α])≤(

Aα|| f ||p

)p

, f ∈ Lp (Ω).

The following lemma involves writing a function as a sum of a functions whose valuesare small and one whose values are large.

Lemma 33.1.2 If p ∈ [1,r], then Lp (Ω)⊆ L1 (Ω)+Lr (Ω).

Proof: Let λ > 0 and let f ∈ Lp (Ω)

f1 (x)≡{

f (x) if | f (x)| ≤ λ

0 if | f (x)|> λ, f2 (x)≡

{f (x) if | f (x)|> λ

0 if | f (x)| ≤ λ.

Thus f (x) = f1 (x)+ f2 (x).∫| f1 (x)|r dµ =

∫[| f |≤λ ]

| f (x)|r dµ ≤ λr−p

∫[| f |≤λ ]

| f (x)|p dµ < ∞.

Therefore, f1 ∈ Lr (Ω).∫| f2 (x)|dµ =

∫[| f |>λ ]

| f (x)|dµ ≤ µ [| f |> λ ]1/p′(∫| f |p dµ

)1/p

< ∞.

This proves the lemma since f = f1 + f2, f1 ∈ Lr and f2 ∈ L1.For f a function having nonnegative real values, α → µ ([ f > α]) is called the distri-

bution function.

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