Beurling transform

The Beurling transform is a mathematical tool used in complex analysis and related fields. It acts on functions defined in the plane and helps analyze their properties, especially in the context of quasiconformal maps and partial differential equations. Think of it as a refined filter that processes a function to reveal or modify certain features, similar to how an equalizer adjusts sound frequencies. Mathematically, it’s a linear operator that transforms functions by integrating them against specific kernels, preserving many properties and playing a key role in understanding deformation and regularity within complex geometric contexts.