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Littlewood–Paley theory

Littlewood–Paley theory uses a decomposition of a function f into a sum of functions f_ρ with localized frequencies. There are several ways to construct such a decomposition; a typical method is as follows.

Iff(x) is a function on R, and ρ is a measurable set (in the frequency space) with characteristic function ${\displaystyle \chi _{\rho }(\xi )}$, then f_ρ is defined via its Fourier transform

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

Informally, f_ρ 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 f_ρ for ρ ∈ Δ.

When Δ consists of the sets of the form

${\displaystyle \rho =[-2^{k+1},-2^{k}]\cup [2^{k},2^{k+1}].}$

for k an integer, this gives a so-called "dyadic decomposition" of f : Σ_ρ f_ρ.

There are many variations of this construction; for example, the characteristic function of a set used in the definition of f_ρ can be replaced by a smoother function.

A key estimate of Littlewood–Paley theory is the Littlewood–Paley theorem, which bounds the size of the functions f_ρ in terms of the size of f. There are many versions of this theorem corresponding to the different ways of decomposing f. A typical estimate is to bound the L^p norm of (Σ_ρ |f_ρ|²)^1/2 by a multiple of the L^p norm of f.

In higher dimensions it is possible to generalize this construction by replacing intervals with rectangles with sides parallel to the coordinate axes. Unfortunately these are rather special sets, which limits the applications to higher dimensions.

The g function is a non-linear operator on L^p(Rⁿ) that can be used to control the L^p norm of a function f in terms of its Poisson integral. The Poisson integral u(x,y) of f is defined for y > 0 by

${\displaystyle u(x,y)=\int _{\mathbb {R} ^{n}}P_{y}($t)f(x-t)\,dt}

where the Poisson kernel P on the upper half space ${\displaystyle \{(y;x)\in \mathbf {R} ^{n+1}\mid y>0\}}$ is given by

${\displaystyle P_{y}($x)=\int _{\mathbb {R} ^{n}}e^{-2\pi it\cdot x-2\pi |t|y}\,dt={\frac {\Gamma ((n+1)/2)}{\pi ^{(n+1)/2}}}{\frac {y}{(|x|^{2}+y^{2})^{(n+1)/2}}}.}

The Littlewood–Paley g function g(f) is defined by

${\displaystyle g($f)(x)=\left(\int _{0}^{\infty }|\nabla u(x,y)|^{2}y\,dy\right)^{1/2}}

A basic property of g is that it approximately preserves norms. More precisely, for 1 < p < ∞, the ratio of the L^p norms of f and g(f) is bounded above and below by fixed positive constants depending on n and p but not on f.

One early application of Littlewood–Paley theory was the proof that if S_n are the partial sums of the Fourier series of a periodic L^p function (p > 1) and n_j is a sequence satisfying n_j+1/n_j > q for some fixed q > 1, then the sequence S_nj converges almost everywhere. This was later superseded by the Carleson–Hunt theorem showing that S_n itself converges almost everywhere.

Littlewood–Paley theory can also be used to prove the Marcinkiewicz multiplier theorem.

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