978 CHAPTER 27. ORLITZ SPACES

This is called the Orlitz class. Also define

LA (Ω)≡ {λu : u ∈ KA (Ω) and λ ∈ F} (27.1.9)

where F is the field of scalars, assumed to be either R or C.The pair (A,Ω) is called ∆ regular if either of the following conditions hold.

A(rx)≤ KrA(x) for all x ∈ [0,∞) (27.1.10)

or µ (Ω)< ∞ and for all r > 0, there exists Mr and Kr > 0 such that

A(rx)≤ KrA(x) for all x≥Mr. (27.1.11)

Note there are N functions which are not ∆ regular. For example, consider

A(x)≡ ex2 −1.

It can’t be ∆ regular because

limr→∞

er2x2 −1ex2 −1

= ∞.

However, functions like xp/p for p > 1 are ∆ regular.Then the following proposition is important.

Proposition 27.1.5 If (A,Ω) is ∆ regular, then KA (Ω) = LA (Ω) . In any case, LA (Ω) is avector space and KA (Ω)⊆ LA (Ω) .

Proof: Suppose (A,Ω) is ∆ regular. Then I claim KA (Ω) is a vector space. This willverify KA (Ω) = LA (Ω) . Let f ,g ∈ KA (Ω) and suppose 27.1.10. Then

A(| f +g|) = A(

2(| f +g|

2

))≤ K2A

(| f +g|

2

)≤ K2

12[A(| f |)+A(|g|)]

so f +g ∈ KA (Ω) in this case. Now suppose 27.1.11∫Ω

A(| f +g|)dµ =∫[| f+g|≤M2]

A(| f +g|)dµ +∫[| f+g|>M2]

A(| f +g|)dµ

≤ A(M2)µ (Ω)+∫

Ω

K2

2(A(| f |)+A(|g|))dµ < ∞.

Thus f +g ∈ KA (Ω) in this case also.Next consider scalar multiplication. First consider the case of 27.1.10. If f ∈ KA (Ω)

and α ∈ F, ∫Ω

A(|α| | f |)dµ ≤ K|α|∫

Ω

A( f )dµ

so in the case of 27.1.10 α f ∈ KA (Ω) whenever f ∈ KA (Ω) . In the case of 27.1.11,∫Ω

A(|α| | f |)dµ =∫[|α|| f |≤M|α|]

A(|α| | f |)dµ +∫[|α|| f |>M|α|]

A(|α| | f |)dµ

≤ A(M|α|

)µ (Ω)+

∫Ω

K|α|A(| f |)dµ < ∞.