Littlewood–Paley decompositions

Choose a smooth radial nonincreasing cutoff $\chi$ on $\mathbb R^d$, equal to one on the unit ball and zero outside the ball of radius two, with values between zero and one. Define $\psi_j(\xi)=\chi(2^{-j}\xi)-\chi(2^{-(j-1)}\xi)$ for integers j. Its support is contained in an annulus with radii $2^{j-1}$ and $2^{j+1}$. Telescoping shows that these nonnegative symbols sum to one at every nonzero frequency. Only a bounded number overlap at any such frequency.

Let $\Delta_jf$ be the Fourier multiplier with symbol $\psi_j$. For L² inputs, finite sums of the bands reconstruct the input in L² as their low and high endpoints tend to opposite infinities, by Plancherel and dominated convergence. Zero frequency is negligible for Euclidean L². On general tempered distributions, a homogeneous construction needs care with polynomials supported at zero in frequency. An inhomogeneous decomposition instead retains one low-frequency block explicitly; its reconstruction and function-space definitions avoid silently throwing away that information.

Define the square function $Sf(x)=(\sum_j|\Delta_jf(x)|^2)^{1/2}$. In L², integrating the square and using Plancherel turns the estimate into comparison of $\sum_j|\psi_j(\xi)|^2$ with one. Finite overlap and a partition summing to one bound that frequency expression above and below by positive constants. This is almost orthogonality: smooth overlapping bands give norm comparison, whereas disjoint orthogonal spectral projections can give exact energy equality.

The Littlewood–Paley theorem extends the two-sided norm comparison to every Euclidean Lᵖ space with exponent strictly between one and infinity for the fixed smooth dyadic family described here. The constants depend on p, dimension, and cutoff bounds. One proof uses uniform estimates for signed sums of the band multipliers and an inequality for random signs, together with duality. Another uses vector-valued singular-integral estimates. Neither proof is a direct pointwise identity, and neither grants the same strong bounds at both endpoints.

A spatial derivative multiplies frequency by its size, so on a band near scale two to the j it contributes roughly that factor. Weighted band energies therefore describe Sobolev regularity and distinguish smooth low-frequency behavior from expensive high-frequency variation. The word roughly is deliberate: an annulus contains a range of magnitudes, so replacing every frequency by the band's nominal scale gives comparable quantities, not exact equality. At the low-frequency end homogeneous norms require their usual interpretation and finiteness assumptions.

This framework is useful when an operator acts differently on different scales or when nonlinear products mix low and high frequencies. Before applying a square-function bound, identify the particular smooth family and verify uniform bounds after dilation. A finite plot of several sine terms can reveal cancellation and show which increment belongs to a frequency band, but it does not prove the infinite-dimensional Lᵖ theorem. Keep the exact finite calculation, the L² comparison proof, and the deeper Lᵖ extension as separate levels of evidence.