Chapter 27

Orlitz Spaces27.1 Basic Theory

All the theorems about the Lp spaces have generalizations to something called an Orlitzspace. [1], [94] Instead of the convex function, A(t) = t p/p, one considers a more generalconvex increasing function called an N function.

Definition 27.1.1 A : [0,∞)→ [0,∞) is an N function if the following two conditions hold.

A is convex and strictly increasing (27.1.1)

limt→0+

A(t)t

= 0, limt→∞

A(t)t

= ∞. (27.1.2)

For A an N function,Ã(s)≡max{st−A(t) : t ≥ 0} . (27.1.3)

As an example see the following picture of a typical N function.

A(t)

Note that from the assumption, 27.1.2 the maximum in the definition of à must exist.This is because for t ̸= 0

(s−A(t)/t) t

is negative for all t large enough. On the other hand, it equals 0 when t = 0 and so it sufficesto consider only t in a compact set.

Lemma 27.1.2 Let φ : R→ R be a convex function. Then φ is Lipschitz continuous on[a,b] .

Proof: Since it is convex, the difference quotients,

φ (t)−φ (a)t−a

are increasing because by convexity, if a < t < x

t−ax−a

φ (x)+(

1− t−ax−a

)φ (a)≥ φ (t)

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