Harmonic Analysis

Harmonic analysis crystallised around Fourier’s idea of decomposing a function into infinite trigonometric series (based on pre-existing tentative works of Bernoulli, Euler, Lagrange), and its continuous analogue, the Fourier transform. This endeavour was motivated by the study of heat and wave equations in certain media (that is, understanding heat diffusion and wave propagation). Fourier’s approach proved fruitful, but several questions regarding convergence were left open and some of them still are.

The “Modern Classical Analysis” is based on similar ideas: depending on the properties of the object to be studied, a decomposition is performed into various “elementary” pieces that are very well understood, and one of the difficulties is to understand how to combine all this information in order to make sense of the initial object. An important example is the Calderon-Zygmund decomposition, which is useful in studying the convolution with the “singular” function $x\mapsto 1/x$ (this is the Hilbert transform). The cancellative property of $1/x$ is captured in the “bad part” of the decomposition, while the “good part” is uniformly bounded.

Other examples are: Carleson’s corona decomposition (for the “Corona Theorem”) and the time-frequency decomposition he introduced for proving the point-wise convergence of Fourier series for $L^p$-integrable functions (for $p>1$); the atomic decomposition which allows to split a function into more integrable pieces while preserving its initial cancellation property, useful to give a simple characterisation of the Hardy space $H^1$ (this space was initially defined as the pre-image of $L^1$ through Riesz transform).