Harmonic Analysis Techniques for Elliptic Operators: Lecture Notes of the 27th Internet Seminar on Evolution Equations (Operator Theory: Advances and Applications, 313)

The study of the Laplacian through the Fourier transform lies at the center of classical harmonic analysis.It is Plancherel’s theorem that intimately links square-integrable functions with the theory of weak derivatives and a symbolic calculus for the Laplacian.Examples include Littlewood-Paley inequalities, Riesz transform estimates and Calderón-Zygmund extrapolation.Over the last decades, the quest to generalize these properties to elliptic operators L in divergence form with bounded measurable coefficients has triggered the development of new techniques that led to a surge of spectacular results in elliptic and parabolic PDE theory. Assuming only undergraduate knowledge in analysis and some background on Hilbert spaces and the Fourier transform, the authors develop the cornerstones of this ‘L-adapted Fourier analysis’ over 14 consecutive lectures.As they delve deeper into the topic, readers make first encounters with maximal functions, Carleson measures and a T(b) theorem.The lectur