Appendix III - The Interpolation Theorems of Marcinkiewicz and Riesz-Thorin

Summary

The simplest example of an interpolation question is as follows. Suppose that the linear operator T is bounded on L¹ and bounded on L². Does it follow that T is bounded on L^p for 1 < p < 2? [The space L^p here is an instance of what is sometimes called an “intermediate space” between L¹ and L².] Note that this question is similar to (but not precisely the same as) one that we faced when considering the L^p boundedness of Calderón-Zygmund singular integral operators. Here we record (special) versions of the Riesz-Thorin Theorem (epitomizing the complex method of interpolation) and the Marcinkiewicz Interpolation Theorem (epitomizing the real method of interpolation) that are adequate for the applications in the present book.

Theorem (Riesz-Thorin):Let 1 ≤ p₀ < p₁ ≤ ∞. Let T be a linear operator on L^p0 ∩ L^p1 such that

and

If 0 ≤ t ≤ 1 and

then we have

Recall that, for 1 ≤ p < ∞, we say that a measurable function ƒ is weak-type p if there is a constant C > 0 such that, for every λ > 0,

We say that ƒ is weak-type ∞ if it is just L^∞. A linear operator T is said to be of weak-type (p, p) if there is a constant C > 0 such that, for each ƒ ∈ L^p and each λ > 0,

An operator is weak-type ∞ if it is simply bounded on L∞ in the classical sense.