Littlewood-Paley Decomposition

In the fields of functional and harmonic analysis, the Littlewood-Paley decomposition is a particular way of decomposing the phase plane which takes a single function and writes it as a superposition of a countably infinite family of functions of varying frequencies. The Littlewood-Paley decomposition is of interest in multiple areas of mathematics and forms the basis for the so-called Littlewood-Paley theory.

To construct the decomposition on , let be a real-valued radial bump function with support

(1)

which equals 1 on the set where

(2)

Next, for , let be a bump function supported on the annulus

(3)

whose derivatives satisfy the inequality

(4)

for some positive number and for all multi-indices . By construction, the bump functions satisfy

(5)

for all , thus providing a specific partition of unity which allows an arbitrary function to be decomposed as

(6)

where is a projection operator (one of the so-called Littlewood-Paley projection operators) defined by

(7)

and where and denote the forward and inverse Fourier transforms in of , respectively. This decomposition for is called its Littlewood-Paley decomposition.

While the above-stated decomposition is for functions assumed to be square integrable, one can decompose likewise nearly any function which has some decay at infinity, e.g., any Schwartz function . For functions , however, the Littlewood-Paley decomposition satisfies a number of important properties. For example, -integrable functions combine with the gradient operator on to satisfy

(8)

and

(9)

facts which imply the heuristic relationship

(10)

and hence that a derivative in can be (heuristically) split into a linear combination of Littlewood-Paley operators. Moreover, by Minkowski's inequality,

(11)

an inequality which combined with the earlier derivative estimates implies the so-called non-endpoint Sobolev embedding inequality:

(12)

for all functions in for which the right-hand side is finite and where satisfies . In the event that and , the above estimates also allow one to prove the so-called endpoint Sobolev embedding inequality:

(13)

for all in for which the right-hand side is finite and where satisfies . These Sobolev embedding inequalities can be expanded even further using fractional differentiation and integration operators to prove the standard Sobolev embedding theorem, a fact which makes the Littlewood-Paley decomposition of particular interest in the study of Sobolev and related spaces.

In practice, the above-stated decomposition is sometimes referred to as the homogeneous Littlewood-Paley decomposition of and is differentiated from a distinct but qualitatively similar decomposition known as the inhomogeneous Littlewood-Paley decomposition. To define the latter, write

(14)

so that

(15)

and redo the above-stated construction so that (rather than ). Though subtle, this distinction leads to the definition of homogeneous and inhomogeneous versions of some function spaces, a separation which is of tantamount importance in a number of contexts.