chapter  8
30 Pages

Lp Classes

If E is ameasurable subset ofRn and p satisfies 0 < p < ∞, then Lp(E) denotes the collection of measurable f for which

E | f |p is finite, that is,

Lp(E) = {

f : E

| f |p < +∞ }

, 0 < p < ∞.

Here, f may be complex-valued (see Exercise 3 of Chapter 4 for the definition of measurability of vector-valued functions). In this case, if f = f1 + if2 for measurable real-valued f1 and f2, we have | f |2 = f 21 + f 22 , so that

∣ f1 ∣

∣ , ∣

∣ f2 ∣

∣ ≤ | f | ≤ ∣∣ f1 ∣

∣ + ∣∣ f2 ∣

∣ .