Littlewood-Paley Decomposition

The Littlewood-Paley decomposition in functional analysis and harmonic analysis 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

which equals 1 on the set where

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

whose derivatives satisfy the inequality

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

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

where is a frequency-localization operator, often called a Littlewood-Paley projection, defined by

The smooth multiplier need not be an idempotent projection operator. Here 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, if the Fourier transform of is supported in , then the gradient satisfies

and

for . These frequency-localized estimates imply the heuristic relationship

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

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

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 inequality of the Sobolev embedding theorem:

for all whose weak gradient belongs to , where satisfies . These inequalities can be expanded even further using fractional differentiation and fractional 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 spaces 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

so that

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 paramount importance in a number of contexts.