986 CHAPTER 27. ORLITZ SPACES

Corollary 27.1.13 Suppose there exists C > 0, a constant such that either

CB(t)≥ A(t)

for all t ≥ 0 orCB(t)≥ A(t)

for all t > M and µ (Ω)< ∞. Then

LB (Ω) ↪→ LA (Ω) .

Proof: If f ∈ LB (Ω) then f = λu where u ∈ KB (Ω) = KCB (Ω) . Hence LCB (Ω) =LB (Ω) and the two norms on LB (Ω) ,

|| ||CB , and || ||B ,

are equivalent norms by the open mapping theorem. Hence by the Proposition 27.1.12, iff ∈ LB (Ω) ,

|| f ||A ≤C1 || f ||CB ≤C2 || f ||Bwhich proves the corollary.

Definition 27.1.14 A increases essentially more slowly than B if for all a > 0,

limt→∞

A(at)B(t)

= 0

The next theorem gives added information on how these spaces are related in case thatone N function increases essentially more slowly than the other.

Theorem 27.1.15 Suppose µ (Ω) < ∞ and A increases essentially more slowly than B.Then

LB (Ω) ↪→ EA (Ω)

Proof: Let f ∈ LB (Ω) . Then there exists λ > 0 such that∫Ω

B(| f |λ

)dµ ≤ 1.

Let r be such that for t ≥ r,A(|λ | t)≤ B(t) .

Then ∫Ω

A(| f |)dµ =∫[| f |≥r]

A(| f |)dµ +∫[| f |<r]

A(| f |)dµ

≤∫

Ω

B(| f ||λ |

)dµ +A(r)µ (Ω)

< 1+A(r)µ (Ω) .