Littlewood–Paley theory

In harmonic analysis, Littlewood–Paley theory is 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 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.

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. 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.

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