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)
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which equals 1 on the set where
(2)
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Next, for , let be a bump function supported on the annulus
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whose derivatives satisfy the inequality
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for some positive number and for all multi-indices . By construction, the bump functions satisfy
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for all , thus providing a specific partition of unity which allows an arbitrary function to be decomposed as
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where is a projection operator (one of the so-called Littlewood-Paley projection operators) defined by
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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
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and
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facts which imply the heuristic relationship
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and hence that a derivative in can be (heuristically) split into a linear combination of Littlewood-Paley operators. Moreover, by Minkowski's inequality,
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an inequality which combined with the earlier derivative estimates implies the so-called non-endpoint Sobolev embedding inequality:
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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:
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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)
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so that
(15)
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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.