Littlewood–Paley theory

In harmonic analysis, a field within mathematics, Littlewood–Paley theory is a theoretical framework used to extend certain results about L² functions to L^p functions for 1 < p < ∞. It is typically used as a substitute for orthogonality arguments which only apply to L^p functions when p = 2. One implementation involves studying a function by decomposing it in terms of functions with localized frequencies, and using the Littlewood–Paley g-function to compare it with its Poisson integral. The 1-variable case was originated by J. E. Littlewood and R. Paley (1931, 1937, 1938) and developed further by Polish mathematicians A. Zygmund and J. Marcinkiewicz in the 1930s using complex function theory (Zygmund 2002, chapters XIV, XV). E. M. Stein later extended the theory to higher dimensions using real variable techniques.