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Riesz-Thorin interpolation theorem

Statement

Assume countable choice. Work with complex-valued $L^{p}$ spaces and a complex-linear operator. Let $(X, A, μ)$ and $(Y, B, ν)$ be measure spaces. Let $1 \leq p_{0}, p_{1} < \infty$ , let $1 < q_{0}, q_{1} < \infty$ , let $0 < θ < 1$ , and define $\frac{1}{p_{θ}} : = \frac{1 - θ}{p_{0}} + \frac{θ}{p_{1}}, \frac{1}{q_{θ}} : = \frac{1 - θ}{q_{0}} + \frac{θ}{q_{1}},$

Suppose $T$ is a complex-linear map from the almost-everywhere classes of complex finite simple functions with finite-measure nonzero set on $X$ to measurable complex almost-everywhere classes on $Y$ , and suppose $M_{0}, M_{1} \geq 0$ satisfy $∥ T f ∥_{q_{0}} \leq M_{0} ∥ f ∥_{p_{0}}, ∥ T f ∥_{q_{1}} \leq M_{1} ∥ f ∥_{p_{1}}$ for every such $f$ . Then $T$ extends uniquely to a bounded linear operator $\tilde{T} : L^{p_{θ}} (μ; C) \to L^{q_{θ}} (ν; C)$ satisfying $∥ \tilde{T} f ∥_{q_{θ}} \leq M_{0}^{1 - θ} M_{1}^{θ} ∥ f ∥_{p_{θ}} (f \in L^{p_{θ}} (μ)) .$

Facts & Assumptions

Given: Countable choice, the complex-linear operator $T$ on finite simple classes, endpoint bounds with nonnegative constants $M_{0}, M_{1}$ , and $0 < θ < 1$ .

[L1]

On arbitrary measure spaces, the finite-simple-core interpolation lemma gives $∥ T f ∥_{q_{θ}} \leq M_{0}^{1 - θ} M_{1}^{θ} ∥ f ∥_{p_{θ}}$ for every complex finite simple class when $q_{0}, q_{1} < \infty$ . Its proof normalizes nonzero inputs and dual tests, establishes finite pairing integrals and strip holomorphy, then uses three-lines and finite-simple dual tests to prove target membership. (Riesz–Thorin estimate on the finite simple core)

[L2]

Complex finite simple classes with finite-measure nonzero set are dense in $L^{p} (μ; C)$ for finite $p$ , on every measure space. (Complex finite-simple and smooth compact-support density for finite p)

[L3]

Assuming countable choice, every complex $L^{q}$ is complete. (Complex Lp completeness and almost-everywhere subsequences, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

technique · the published normalized finite-simple-core estimate, followed by completion

1.1L1given

Since both target endpoint exponents are finite, [L1] applies on the arbitrary measure spaces in the statement. Put $K = M_{0}^{1 - θ} M_{1}^{θ}$ . For every finite simple class $f$ in the domain of $T$ , it proves both $T f \in L^{q_{θ}} (ν; C)$ and [L1, given] $∥ T f ∥_{q_{θ}} \leq K ∥ f ∥_{p_{θ}} .$ The source proof treats $f = 0$ separately and normalizes every nonzero $f$ before applying the three-lines bound, so the endpoint powers of its norm are not replaced by a first power without justification. This also covers $M_{0} = 0$ or $M_{1} = 0$ .

2.1L2L3step 1.1

Let $f \in L^{p_{θ}} (μ; C)$ . Since $p_{θ} < \infty$ , [L2] supplies a sequence of finite simple classes $f_{n} \to f$ in $L^{p_{θ}}$ . By step 1.1, $(T f_{n})$ is Cauchy in $L^{q_{θ}}$ . Countable choice is used through [L3] to obtain its limit; define $\tilde{T} f = \lim_{n} T f_{n}$ .

3.1step 1.1step 2.1

If $g_{n} \to f$ is another finite-simple approximation, step 1.1 gives [step 1.1, step 2.1] $∥ T f_{n} - T g_{n} ∥_{q_{θ}} \leq K ∥ f_{n} - g_{n} ∥_{p_{θ}} \to 0,$ so the limit is independent of the approximation. Taking limits in the same estimate gives $∥ \tilde{T} f ∥_{q_{θ}} \leq K ∥ f ∥_{p_{θ}}$ . Constant approximations show that $\tilde{T}$ extends $T$ , and termwise linearity shows that it is complex-linear.

4.1L2step 3.1∎

Any bounded linear extension agrees with $T$ on the dense core from [L2]; continuity makes it agree with $\tilde{T}$ on every $f$ . Thus the extension is unique and has the asserted norm bound.

Depends on

Used by

The Marcel Riesz conjugate-function theorem on the circle Theorem

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Theorem 6.27 (standard reference, not scraped)
Richard F. Bass, Real Analysis for Graduate Students, Chapter 24.2 (standard reference, not scraped)