E. M. Stein later extended the theory to higher dimensions using real variable techniques.
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 fρ for ρ ∈ Δ.
When Δ consists of the sets of the form for k an integer, this gives a so-called "dyadic decomposition" of f : Σρ fρ.
A key estimate of Littlewood–Paley theory is the Littlewood–Paley theorem, which bounds the size of the functions fρ 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 Lp norm of (Σρ |fρ|2)1/2 by a multiple of the Lp 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.
More precisely, for 1 < p < ∞, the ratio of the Lp 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 Sn are the partial sums of the Fourier series of a periodic Lp function (p > 1) and nj is a sequence satisfying nj+1/nj > q for some fixed q > 1, then the sequence Snj converges almost everywhere.