Orlicz Spaces and Generalized Orlicz Spaces by Petteri Harjulehto & Peter Hästö

Author:Petteri Harjulehto & Peter Hästö
Language: eng
Format: epub
ISBN: 9783030151003
Publisher: Springer International Publishing

Then and . It follows from (A1) for each ball B j that and hence

If r < 1, then since φ −1 is increasing, and (aDec)1 of φ −1 (Proposition 2.​3.​7) gives . Combining this with the previous inequality, we see that the condition holds. If , then increasing and (aDec)1 are applied the other way around to reach the same conclusion.

Analogous steps also show that the condition implies (A1), so the proof is complete. □

4.2 The Decay Condition (A2)

Definition 4.2.1

We say that φ ∈ Φw( Ω) satisfies (A2) if for every s > 0 there exist β ∈ (0, 1] and h ∈ L 1( Ω) ∩ L ∞( Ω) such that

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