2026/02/23 by Asuman Güven Aksoy, Daniel Akech Thiong
#math.AP
This article explores the vital role of interpolation theory and Lorentz spaces in the rigorous analysis of linear operators. While classical Lebesgue spaces (Lp) successfully measure the magnitude of functions, they frequently fail to bound evolution operators at critical endpoints of p=1 or p = ∞ because they conflate a function's amplitude with its spatial spread. To resolve this analytic bottleneck, we introduce distribution functions and decreasing rearrangements, culminating in the construction of Lorentz spaces (Lp, q). By utilizing the Complex (Riesz-Thorin), Real (Peetre's K-functional), and Marcinkiewicz methods of interpolation, these highly sensitive intermediate spaces act as geometric bridges between endpoint extremes. We conclude by applying this abstract framework to two distinct illustrative models: deriving the continuous smoothing decay of the parabolic Heat equation, and establishing the foundational dispersive Strichartz estimates for the dispersive free Schrödinger equation.