Littlewood, Paley and Almost-Orthogonality: a theory well ahead of its time (2511.22605v1)

Abstract: Littlewood--Paley theory began with the classic paper of Littlewood and Paley (J.\ E.\ Littlewood, R.\ E.\ A.\ C.\ Paley, {\em Theorems on Fourier Series and Power Series}. J. Lond. Math. Soc. (1), {\bf 6} (1931), 230--33). We discuss this paper and its impact from a historical perspective. We include an outline of the results in the paper and their subsequent significance in relation to developments over the last century, and set them into the context of the current state of the art in harmonic analysis and beyond.

• The paper outlines Littlewood–Paley theory's extension of classical L² orthogonality to Lᵖ spaces using quadratic square functions and dyadic decompositions.
• It presents analytic techniques including Fourier multipliers, maximal function inequalities, and the role of almost-orthogonality in bridging gaps in harmonic analysis.
• The discussion connects historical developments to modern applications such as wavelet analysis, PDEs, and decoupling estimates in higher dimensions.

Littlewood–Paley and Almost-Orthogonality: Theoretical Foundations and Evolution

Introduction and Context

The paper “Littlewood, Paley and Almost-Orthogonality: a theory well ahead of its time” (2511.22605) by Anthony Carbery offers a comprehensive, historically-contingent analysis of the inception, development, and far-reaching influence of Littlewood–Paley theory. The work positions the subject within the trajectory of modern harmonic analysis, emphasizing its role as the archetype for almost-orthogonality phenomena and bridging Hilbert space (orthogonal) techniques into the more general, and technically challenging, $L^p$-framework with $p \ne 2$.

Foundational Contributions of Littlewood–Paley Theory

The Littlewood–Paley theory originated from the pioneering work of Littlewood and Paley [LP1, 1931], who posed the crucial question: to what extent can orthogonality-inspired methods be extended to $L^p$ spaces for $p \ne 2$? By constructing Fourier multipliers $Q_j$ associated with dyadic decompositions in frequency space, they proved the pivotal inequality: for $1<p<\infty$, there exist constants $A_p$, $B_p$ such that

$$A_p\|f\|_p \le \left\| \left(\sum_{j} |Q_j f|^2 \right)^{1/2} \right\|_p \le B_p \|f\|_p$$

This result introduced the square function as a substitute for the classical orthogonality relation in $L^2$ and established “almost-orthogonality” as a viable analytic tool in $L^p$ settings, where direct use of inner products is unavailable. The quadratic expression $\mathcal{Q}f = (\sum_j |Q_j f|^2)^{1/2}$ is central to this methodology.

Among the immediate corollaries, Littlewood and Paley produced a multiplier theorem for sequences $\{r_j\}$ with $|r_j|\le1$, asserting boundedness of $\sum_j r_j Q_j$ on $L^p$ for $1<p<\infty$. This result is a precursor to later general multiplier theorems (Marcinkiewicz, Hörmander–Mikhlin), highlighting the power of uniform control over Fourier multipliers when paired with almost-orthogonality principles.

Maximal Functions, Pointwise Convergence, and Lacunary Series

The original paper also addresses questions of pointwise a.e. convergence of Fourier series and their lacunary subsequences, foundational issues in analysis with connections to the Hardy–Littlewood maximal function. Littlewood and Paley established maximal inequalities for lacunary partial sums, for $1<p\leq2$, yielding strong $L^p$-norm control and implications for almost-everywhere convergence. The quadratic mediation between pointwise maximal functions and $L^p$-norms provided a paradigm for subsequent research into differentiation theory and maximal inequalities in higher dimensions.

Real Variable Methods and the Calderón–Zygmund Revolution

While the theory’s initial proofs relied on function theory in one complex variable, the advent of real-variable methods, especially via Calderón–Zygmund theory (singular integrals) and Stein’s vector-valued extensions, released the full potential of Littlewood–Paley techniques. Square function estimates in general settings—$\mathbb{R}^n$, locally compact abelian groups, and more abstract geometric environments—became accessible, enabling the characterization and study of differentiability, function spaces (Sobolev, Bessel, Triebel–Lizorkin, Besov) and the $L^p$ mapping behavior of a wide class of singular integral operators.

Vector-valued inequalities and weighted norm inequalities amplified the flexibility of almost-orthogonality, underpinning results not just in scalar-valued harmonic analysis, but also in Banach-space-valued and UMD settings [see, e.g., Bourgain 1983, McConnell 1984].

Quadratic Expressions in Probability, Ergodic Theory, and Semigroups

Littlewood–Paley theory's abstraction was furthered by its parallel development in probability, particularly martingale theory. Results such as the Burkholder–Davis–Gundy inequalities for martingale square functions demonstrated the equivalence between $L^p$-norms of a martingale and its difference sequence, subsuming Marcinkiewicz–Zygmund inequalities for independent mean-zero random variables. This abstraction facilitated the transference of almost-orthogonality and square function methods to ergodic theory, symmetric diffusion semigroups, and stochastic analysis.

Wavelets, Paraproducts, and the Calderón Reproducing Formula

The emergence of wavelet theory, rooted in the Calderón reproducing formula, connected square functions to quasidiagonalization of operators and unconditional decompositions in $L^p$ spaces. Wavelet coefficients provide a robust framework for both $L^2$ and $L^p$ analysis, enabling precise representation theorems, Littlewood–Paley-type inequalities, and efficient computation. Paraproducts, arising in the context of non-linear PDEs and fractional derivate rules, exploit these decompositions and have led to a rich harmonic analysis of product-like operators.

Exotic Square Functions and Almost-Orthogonality Beyond Dyadic Decompositions

The theoretical boundaries of Littlewood–Paley theory were extended by investigations into square function inequalities indexed by non-dyadic, equally-spaced, or even arbitrary disjoint intervals (Carleson, Rubio de Francia). Notably, Rubio de Francia’s theorem established that for arbitrary intervals, the square function $\mathcal{Q}f$ satisfies

$$\|\mathcal{Q}f\|_p \lesssim \|f\|_p, \quad 2 \leq p < \infty$$

with constants independent of the choice of intervals. This highly nontrivial result harnesses vector-valued Calderón–Zygmund theory even in settings devoid of dilation symmetry.

Angular Littlewood–Paley decompositions, especially connected to curvature and Kakeya set phenomena, demonstrate both the reach and the limits of the theory, directly encoding results (and counterexamples) in harmonic analysis, such as Fefferman’s disc multiplier theorem and decoupling for oscillatory integral operators.

Geometric Measure Theory, PDE, and Recent Developments

Quadratic expressions have become intrinsic in the solution of major problems in geometric measure theory and PDE. Notably, advances in boundary value problems for elliptic and parabolic operators, the Kato square root problem, and the connection to rectifiability (via Jones’ $\beta$-numbers) rely on the analytic control conferred by square functions. The theory facilitates quantitative and qualitative breakthroughs in understanding fine geometric and analytic structure.

Connections to Open Problems and Current Research

The ramifications of almost-orthogonality permeate several central open problems: Kakeya-type maximal inequalities, boundedness of Bochner–Riesz means, and the restriction problem. Sharp reverse Littlewood–Paley inequalities, such as

$$\left\| \sum_\nu S_\nu f \right\|_{2n/(n-1)} \lesssim \left\| \left( \sum_\nu |S_\nu f|^2 \right)^{1/2} \right\|_{2n/(n-1)}$$

remain open in higher dimensions and, if resolved, would provide unified solutions to these core questions. The recent resolution of related estimates for the cone in $\mathbb{R}^3$ [Guth–Wang–Zhang, Ann. of Math. 2020] underscores ongoing developments.

Decoupling estimates, a modern outgrowth of square function techniques, have led to applications in analytic number theory (e.g., Vinogradov’s mean value theorem [Bourgain–Demeter–Guth]), local smoothing, and the geometry of solutions to dispersive PDEs.

Conclusion

Littlewood–Paley theory, as depicted in Carbery’s survey (2511.22605), is the archetype of a mathematical framework whose abstraction, flexibility, and depth have generated entire subfields in modern analysis. Its techniques, especially the utilization of quadratic expressions to realize almost-orthogonality in the absence of true orthogonality, are pervasive across harmonic analysis, PDE, probability, and geometry. While the theory’s classical inequalities and constructions remain foundational, its evolution into ever more abstract and geometric settings continues to drive current and future research. The structural insights it affords are likely to remain indispensable for further advances in areas ranging from decoupling theory and geometric measure theory to operator analysis in high dimensions.