Littlewood–Paley inequality

Remarks

For $p = 2$ the inequality follows from Plancherel's theorem. For the upper bound, view $f \mapsto (\Delta_j f)_{j \in \Z}$ as a single operator from $L^p(\R^n)$ to $L^p(\R^n; \ell^2)$. Its kernel $(\check\psi_j)_j$ satisfies the size and smoothness estimates of a vector-valued Calderón–Zygmund operator, so the $L^2$ bound extends to $1 < p < \infty$. The lower bound follows by duality. Let $\tilde\Delta_j = \Delta_{j-1} + \Delta_j + \Delta_{j+1}$; its multiplier equals $1$ on the support of the multiplier of $\Delta_j$, so $\int f \bar g = \sum_j \int \Delta_j f \, \overline{\tilde\Delta_j g}$ for nice $f$ and $g$. The Cauchy–Schwarz inequality in $j$ and Hölder's inequality give $\abs{\int f \bar g} \le \norm{Sf}_p \norm{\tilde S g}_{p'}$, and the upper bound for the square function $\tilde S$ of the $\tilde\Delta_j$ in $L^{p'}$ finishes the proof.

The inequality fails for $p = 1$ and $p = \infty$. At $p = 1$ the upper bound fails already for an integrable $f$ with $\int f \neq 0$, and the correct substitute replaces $L^1$ by the real Hardy space $H^1$.

Going deeper

For a student

A typical consequence: for any choice of signs $\varepsilon_j \in \set{-1, 1}$, the operator $f \mapsto \sum_j \varepsilon_j \Delta_j f$ is bounded on $L^p$ with a bound independent of the signs. This is the first step towards the Marcinkiewicz and Hörmander–Mikhlin multiplier theorems, where a multiplier is cut into dyadic pieces and each piece is handled separately. The inequality is also the starting point for the characterization of Sobolev spaces $W^{k,p}$, Besov spaces and Triebel–Lizorkin spaces through the sizes of the $\Delta_j f$. On the torus, the analogous statement with sharp dyadic blocks of the Fourier series is the original theorem of Littlewood and Paley.