Littlewood–Paley theory

In harmonic analysis, a field within mathematics, Littlewood–Paley theory is a theoretical framework used to extend certain results about square-integrable 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 R. 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(x) is a function on R, and ρ is a measurable set (in the frequency space) with characteristic function

χ

ρ

(

ξ

)

{\displaystyle \chi _{\rho }(\xi )}

, then fρ is defined via its Fourier transform

f

^

ρ

:=

χ

ρ

f

^

{\displaystyle {\hat {f}}_{\rho }:=\chi _{\rho }{\hat {f}}}

.

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 on 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

ρ

=

[

−

2

k

+

1

,

−

2

k

]

∪

[

2

k

,

2

k

+

1

]

.