33.1. THE MARCINKIEWICZ INTERPOLATION THEOREM 1121

Proof of the claim: [ fn > α] ↑ [ f > α] because if f (x) > α then for large enough n,fn (x)> α and so

µ ([ fn > α]) ↑ µ ([ f > α]).

This proves the lemma. (Note the importance of the strict inequality in [ f > α] in provingthe claim.)

The next theorem is the main result in this section. It is called the Marcinkiewiczinterpolation theorem.

Theorem 33.1.4 Let (Ω,µ,S ) be a σ finite measure space, 1 < r < ∞, and let

T : L1 (Ω)+Lr (Ω)→ space of measurable functions

be subadditive, weak (r,r), and weak (1,1). Then T is of type (p, p) for every p ∈ (1,r)and

||T f ||p ≤ Ap || f ||pwhere the constant Ap depends only on p and the constants in the definition of weak (1,1)and weak (r,r).

Proof: Let α > 0 and let f1 and f2 be defined as in Lemma 33.1.2,

f1 (x)≡{

f (x) if | f (x)| ≤ α

0 if | f (x)|> α, f2 (x)≡

{f (x) if | f (x)|> α

0 if | f (x)| ≤ α.

Thus f = f1 + f2 where f1 ∈ Lr and f2 ∈ L1. Since T is subadditive ,

[|T f |> α]⊆ [|T f1|> α/2]∪ [|T f2|> α/2] .

Let p ∈ (1,r). By Lemma 33.1.3,∫|T f |p dµ ≤ p

∫∞

0α

p−1µ ([|T f1|> α/2])dα+

+p∫

∞

0α

p−1µ ([|T f2|> α/2])dα.

Therefore, since T is weak (1,1) and weak (r,r),∫|T f |p dµ ≤ p

∫∞

0α

p−1(

2Ar

α|| f1||r

)r

dα + p∫

∞

0α

p−1 2A1

α|| f2||1 dα. (33.1.2)

Therefore, the right side of 33.1.2 equals

p(2Ar)r∫

∞

0α

p−1−r∫

Ω

| f1|r dµdα +2A1 p∫

∞

0α

p−2∫

Ω

| f2|dµdα =

p(2Ar)r∫

Ω

∫∞

0α

p−1−r | f1|r dαdµ +2A1 p∫

Ω

∫∞

0α

p−2 | f2|dαdµ.