Besov

Besov spaces, named after Oleg V. Besov, are a scale of function spaces used in harmonic analysis, function approximation, and PDE theory. They generalize both Sobolev and Hölder spaces and can be defined on R^n, domains in R^n, or manifolds. The standard notation is B^s_{p,q}, where s is a smoothness parameter, p controls integrability, and q governs summability across scales. There are several equivalent definitions, including through moduli of smoothness, Littlewood-Paley decompositions with dyadic frequency pieces, and wavelet coefficient characterizations.

The parameter s measures how smooth a function is, while p and q describe how the size of the function and its variations are aggregated across scales. Besov spaces form a natural interpolation scale between different function spaces and interpolate well between Sobolev and Hölder spaces. Special cases include B^m_{p,p}, which coincides with the classical Sobolev space W^{m,p} for nonnegative integers m; for 0<s<1, B^s_{p,p} is closely related to fractional Sobolev spaces.

Besov spaces exhibit sharp embedding and trace properties that depend on the dimension, s, p, and q, making them well suited to questions about regularity and boundary behavior. They also admit robust wavelet characterizations, which facilitate numerical methods and practical computations. In broader contexts, Besov spaces form part of the Besov–Triebel–Lizorkin scale, a comprehensive framework that extends the concept to a wider range of smoothness notions and parameter choices. Besov spaces are foundational in analysis and have wide-ranging applications in PDEs, signal processing, and approximation theory.