mathematics

Littlewood subordination theorem

Littlewood subordination theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Littlewood subordination theorem rather than just read about it. In short: In mathematics, the Littlewood subordination theorem, proved by J. E.

Key takeaways

Littlewood subordination theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Littlewood subordination theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Littlewood subordination theorem from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Littlewood subordination theorem, proved by J. E. Littlewood in 1925, is a theorem in operator theory and complex analysis. It states that any holomorphic univalent self-mapping of the unit disk in the complex numbers that fixes 0 induces a contractive composition operator on various function spaces of holomorphic functions on the disk. These spaces include the Hardy spaces, the Bergman spaces and Dirichlet space.

Subordination theorem Let h be a holomorphic univalent mapping of the unit disk D into itself such that h(0) = 0. Then the composition operator Ch defined on holomorphic functions f on D by

C h ( f ) = f ∘ h {\displaystyle C_{h}(f)=f\circ h}

defines a linear operator with operator norm less than 1 on the Hardy spaces H p ( D ) {\displaystyle H^{p}(D)} , the Bergman spaces A p ( D ) {\displaystyle A^{p}(D)} . (1 ≤ p < ∞) and the Dirichlet space D ( D ) {\displaystyle {\mathcal {D}}(D)} . The norms on these spaces are defined by:

‖ f ‖ H p p = sup r 1 2 π ∫ 0 2 π | f ( r e i θ ) | p d θ {\displaystyle \|f\|_{H^{p}}^{p}=\sup _{r}{1 \over 2\pi }\int _{0}^{2\pi }|f(re^{i\theta })|^{p}\,d\theta }

‖ f ‖ A p p = 1 π ∬ D | f ( z ) | p d x d y {\displaystyle \|f\|_{A^{p}}^{p}={1 \over \pi }\iint _{D}|f(z)|^{p}\,dx\,dy}

‖ f ‖ D 2 = 1 π ∬ D | f ′ ( z ) | 2 d x d y = 1 4 π ∬ D | ∂ x f | 2 + | ∂ y f | 2 d x d y {\displaystyle \|f\|_{\mathcal {D}}^{2}={1 \over \pi }\iint _{D}|f^{\prime }(z)|^{2}\,dx\,dy={1 \over 4\pi }\iint _{D}|\partial _{x}f|^{2}+|\partial _{y}f|^{2}\,dx\,dy}

Littlewood's inequalities Let f be a holomorphic function on the unit disk D and let h be a holomorphic univalent mapping of D into itself with h(0) = 0. Then if 0 < r < 1 and 1 ≤ p < ∞

∫ 0 2 π | f ( h ( r e i θ ) ) | p d θ ≤ ∫ 0 2 π | f ( r e i θ ) | p d θ . {\displaystyle \int _{0}^{2\pi }|f(h(re^{i\theta }))|^{p}\,d\theta \leq \int _{0}^{2\pi }|f(re^{i\theta })|^{p}\,d\theta .}

This inequality also holds for 0 < p < 1, although in this case there is no operator interpretation.

Proofs

Case p = 2 To prove the result for H2 it suffices to show that for f a polynomial

‖ C h f ‖ 2 ≤ ‖ f ‖ 2 , {\displaystyle \displaystyle {\|C_{h}f\|^{2}\leq \|f\|^{2},}}

Let U be the unilateral shift defined by

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Littlewood subordination theorem

Start with the simplest possible case. Write down what Littlewood subordination theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Littlewood subordination theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Littlewood subordination theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Littlewood subordination theorem

In research: Littlewood subordination theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Littlewood subordination theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Littlewood subordination theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Operator theory, Theorems in complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Littlewood subordination theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Littlewood subordination theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Littlewood subordination theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Littlewood subordination theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Littlewood subordination theorem in simple terms?

In mathematics, the Littlewood subordination theorem, proved by J. E.

Why does Littlewood subordination theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Littlewood subordination theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Littlewood subordination theorem.