Overview
Besov spaces form a scale of Banach (and quasi-Banach in some ranges) spaces that quantify the smoothness of functions or distributions in a flexible way. They are indexed by three parameters: a smoothness exponent s, an integrability exponent p, and a summability exponent q. Compared with classical Sobolev or Hölder spaces, Besov spaces capture fractional and anisotropic regularity and are well suited to fine regularity questions.
Definitions and characterizations
There are several equivalent ways to define Besov spaces. One is by using finite differences: control of k-th order difference increments at scale h measured in L^p and summed with an l^q-norm across scales defines the space B^s_{p,q}. Another common approach uses a Littlewood–Paley frequency decomposition: the size of dyadic frequency pieces weighted by 2^{js} and aggregated in l^q produces an equivalent norm. Wavelet coefficient decay also provides a discrete characterization, which is convenient in numerical work.
Relations and notable facts
Besov spaces generalize several classical spaces. For example, B^s_{2,2} coincides with the L^2-based Sobolev (or Bessel potential) space H^s. When q=\infty and p=\infty, certain Besov spaces recover Hölder–Zygmund classes. Besov scales enjoy embedding theorems, interpolation properties, and trace theorems specifying boundary values of functions on lower-dimensional sets.
Applications and importance
These spaces appear across analysis and applied mathematics: they provide natural regularity frameworks for nonlinear partial differential equations, for quantifying error in approximation and compression, and in image processing where sparsity in wavelet bases is modeled by Besov norms. For introductions focused on PDE applications see PDE theory resources.
History and context
The spaces are named after the Russian mathematician Oleg Besov who studied them in the mid-20th century. Modern theory connects them to interpolation theory, Fourier analysis, and harmonic analysis, making them a central tool when fractional smoothness and fine-scale structure matter.