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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 R^n, let psi(xi) be a real-valued radial bump function with support

 supp(psi(xi))={xi in R^n:||xi||<=2}
(1)

which equals 1 on the set I where

 I={xi in R^n:||xi||<=1}.
(2)

Next, for j in Z, let phi_j(xi)=psi(2^(-j)xi)-psi(2^(-j+1)xi) be a bump function supported on the annulus

 A_phi={1/2<=||xi||<=2}
(3)

whose derivatives satisfy the inequality

 2^(j|alpha|)|partial^alphaphi_j(xi)|<=c_alpha
(4)

for some positive number c_alpha and for all multi-indices alpha in (Z^*)^n. By construction, the bump functions phi_j satisfy

 sum_(j in Z)phi_j=1
(5)

for all xi!=0, thus providing a specific partition of unity which allows an arbitrary function f in L^2=L^2(R^n,dx^n) to be decomposed as

 f=sum_(j in Z)P_jf,
(6)

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

 P_j(f)=F^(-1)(phi_jF[f](xi))
(7)

and where F[f](x) and F^(-1)[f](x) denote the forward and inverse Fourier transforms in R^n of f, respectively. This decomposition for f is called its Littlewood-Paley decomposition.

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

 ||del f||_p∼2^j||f||_p
(8)

and

 ||del P_jf||_p∼2^j||P_jf||_p,
(9)

facts which imply the heuristic relationship

 del ∼sum_(j in Z)2^jP_j
(10)

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

 sup_(j in Z)||P_jf||_p<~||f||_p<~sum_(j in Z)||P_jf||_p,
(11)

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

 ||f||_(L^q(R^n))<~||f||_(L^p(R^n))+||del f||_(L^p(R^n))
(12)

for all functions f in R^n for which the right-hand side is finite and where 1<=p<q<=infty satisfies 1/p-1/n>1/q. In the event that p!=1 and q!=infty, the above estimates also allow one to prove the so-called endpoint Sobolev embedding inequality:

 ||f||_(L^q(R^n))<~||del ||_(L^p(R^n))
(13)

for all f in R^n for which the right-hand side is finite and where 1<p<q<infty satisfies 1/p-1/n=1/q. 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 f and is differentiated from a distinct but qualitatively similar decomposition known as the inhomogeneous Littlewood-Paley decomposition. To define the latter, write

 phi_0(xi)=psi(xi)
(14)

so that

 supp(phi_0) subset {xi in R^n:||xi||<=2}
(15)

and redo the above-stated construction so that j in Z^* (rather than j in Z). 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.


See also

Fourier Transform, Fractional Derivative, Fractional Integral, Harmonic Analysis, Homogeneous Littlewood-Paley Decomposition, Inhomogeneous Littlewood-Paley Decomposition, L-p-Space, Phase Plane, Schwartz Function, Sobolev Space

Portions of this entry contributed by Christopher Stover

Portions of this entry contributed by Lin Cong

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References

Runst, T. and Sickel, W. Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Berlin, Germany: de Gruyter, 1996.Tao, T. "Lecture Notes 2 for 254A." https://www.math.ucla.edu/~tao/254a.1.01w/notes2.ps.

Cite this as:

Cong, Lin; Stover, Christopher; and Weisstein, Eric W. "Littlewood-Paley Decomposition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Littlewood-PaleyDecomposition.html

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