World Library  
Flag as Inappropriate
Email this Article

Littlewood–Paley theory

Article Id: WHEBN0024465861
Reproduction Date:

Title: Littlewood–Paley theory  
Author: World Heritage Encyclopedia
Language: English
Subject: April 1933, Theory
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Littlewood–Paley theory

In harmonic analysis, Littlewood–Paley theory is a theoretical framework used to extend certain results about L2 functions to Lp functions for 1<p<∞. It is typically used as a substitute for orthogonality arguments which only apply to Lp functions when p=2. One implementation involves studying a function by decomposing it in terms of functions with localized frequencies, and using the Littlewood–Paley g-function to compare it with its Poisson integral. The 1-variable case was originated by J. E. Littlewood and Raymond Paley (1931, 1937, 1938) and developed further by Polish mathematicians A. Zygmund and J. Marcinkiewicz in the 1930s using complex function theory (Zygmund 2002, chapters XIV, XV). E. M. Stein later extended the theory to higher dimensions using real variable techniques.

The dyadic decomposition of a function

Littlewood–Paley theory uses a decomposition of a function f into a sum of functions fρ with localized frequencies. There are several ways to construct such a decomposition; a typical method is as follows.

If f is a function on R, and ρ is a measurable set with characteristic function χρ, then fρ is defined to be given by

\hat f_\rho = \chi_\rho\hat f

where the "hat" is used to represent the Fourier transform. Informally, fρ is the piece of f whose frequencies lie in ρ.

If Δ is a collection of measurable sets which (up to measure 0) are disjoint and have union the real line, then a well behaved function f can be written as a sum of functions fρ for ρ ∈Δ.

When Δ consists of the sets of the form

\rho = [-2^{k+1},-2^k] \cup [2^k,2^{k+1}].

for k an integer, this gives a so-called "dyadic decomposition" of f: Σρ fρ.

There are many variations of this construction; for example, the characteristic function of a set used in the definition of fρ can be replaced by a smoother function.

A key estimate of Littlewood–Paley theory is the Littlewood–Paley theorem, which bounds the size of the functions fρ in terms of the size of f. There are many versions of this theorem corresponding to the different ways of decomposing f. A typical estimate is to bound the Lp norm of (Σρ |fρ|2)1/2 by a multiple of the Lp norm of f.

In higher dimensions it is possible to generalize this construction by replacing intervals with rectangles with sides parallel to the coordinate axes. Unfortunately these are rather special sets, which limits the applications to higher dimensions.

The Littlewood–Paley g function

The g function is a non-linear operator on Lp(Rn) that can be used to control the Lp norm of a function f in terms of its Poisson integral. The Poisson integral u(x,y) of f is defined for y>0 by

u(x,y) = \int_{R^n}P_y(t)f(x-t)dt

where the Poisson kernel P is given by

P_y(x) = \int_{R^n}e^{-2\pi i tx-2\pi |t|y}dt = \frac{\Gamma((n+1)/2)}{\pi^{(n+1)/2}}\frac{y}{(|x|^2+y^2)^{(n+1)/2}}

The Littlewood–Paley g function g(f) is defined by

g(f)(x) = (\int_0^\infty|\nabla u(x,y)|^2ydy)^{\frac{1}{2}}

A basic property of g is that it approximately preserves norms. More precisely, for 1<p<∞, the ratio of the Lp norms of f and g(f) is bounded above and below by fixed positive constants depending on n and p but not on f.

Applications

One early application of Littlewood–Paley theory was the proof that if Sn are the partial sums of the Fourier series of a periodic Lp function (p>1) and nj is a sequence satisfying nj+1/nj > q for some fixed q>1, then the sequence Snj converges almost everywhere. This was later superseded by the Carleson–Hunt theorem showing that Sn itself converges almost everywhere.

Littlewood–Paley theory can also be used to prove the Marcinkiewicz multiplier theorem.

References

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.