In harmonic analysis, Littlewood–Paley theory is a term used to describe a theoretical framework used to extend certain results about L² functions to L^p functions for 1<p<∞. It is typically used as a substitute for orthogonality arguments which only apply to L^p 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 and developed further by Zygmund and Marcinkiewicz in the 1930s using complex function theory . 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 f_ρ 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 χ_ρ, then f_ρ is defined to be given by
where the "hat" is used to represent the Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

. Informally, f_ρ 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 f_ρ
for ρ ∈Δ.

When Δ consists of the sets of the form.
for k an integer, this gives a so-called "dyadic decomposition" of f: Σ_ρ f_ρ.

There are many variations of this construction; for example, the characteristic function of a set used in the definition of f_ρ 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 f_ρ 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 L^p norm of (Σ_ρ |f_ρ|²)^1/2 by a multiple of the L^p 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 L^p(Rⁿ) that can be used to control the L^p 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
where the Poisson kernel

Poisson kernel

In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation...

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

A basic property of g is that it approximately preserves norms. More precisely, for 1<p<∞, the ratio of the L^p 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 S_n are the partial sums of the Fourier series of a periodic L^p function (p>1) and n_j is a sequence satisfying n_j+1/n_j > q for some fixed q>1, then the sequence S_nj converges almost everywhere. This was later superseded by the Carleson–Hunt theorem

Carleson's theorem

Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise almost everywhere convergence of Fourier series of L2 functions, proved by...

 showing that S_n itself converges almost everywhere.

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