Fourier analysis has been an extraordinarily powerful mathematical tool since its development 200 years ago, and currently has a wide range of applications in diverse scientific fields including digital image processing, forensics, option pricing, cryptography, optics, oceanography, and protein structure analysis. Like a prism that decomposes a beam of light into a rainbow of colors, Fourier analysis transforms signals into a mathematical spectrum of basic wave components of different amplitudes and frequencies, from which many hidden properties in the data can be deciphered. At the abstract mathematical level signals are represented by functions and their filtering and other operations on them by operators. From a functional analytical point of view, these objects are studied by decomposing them into elementary building blocks, some of which have wavelike behavior too. Decomposition techniques such as atomic, molecular, wavelet and wave-packet expansions provide a multi-scale refinement of Fourier analysis and exploit a rather simple concept: "waves with very different frequencies are almost invisible to each other". Many of these useful techniques have been developed around the study of some particular operators called singular integral operators and, recently, similar techniques have been pushed to the analysis of new multilinear operators that arise in the study of (para) product-like operations, null-forms, and other nonlinear functional expressions. In this talk we will present some of our contributions in the study of multilinear singular integrals and function spaces, and their applications to the development of the equivalent of the calculus Leibniz rule to the concept of fractional derivatives.
