Besov space
# Besov space

In mathematics, the Besov space (named after Oleg Vladimirovich Besov) $B_{p,q}^{s}(\mathbf {R} )$ is a complete quasinormed space which is a Banach space when 1 ≤ p, q ≤ ∞. These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize more elementary function spaces such as Sobolev spaces and are effective at measuring regularity properties of functions.

## Definition

Several equivalent definitions exist. One of them is given below.

Let

    $\Delta _{h}f(x)=f(x-h)-f(x)$

and define the modulus of continuity by

    $\omega _{p}^{2}(f,t)=\sup _{|h|\leq t}\left\|\Delta _{h}^{2}f\right\|_{p}$

Let n be a non-negative integer and define: s = n + α with 0 < α ≤ 1. The Besov space $B_{p,q}^{s}(\mathbf {R} )$ contains all functions f such that

    $f\in W^{n,p}(\mathbf {R} ),\qquad \int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}\left(f^{(n)},t\right)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}<\infty .$

## Norm

The Besov space $B_{p,q}^{s}(\mathbf {R} )$ is equipped with the norm

    $\left\|f\right\|_{B_{p,q}^{s}(\mathbf {R} )}=\left(\|f\|_{W^{n,p}(\mathbf {R} )}^{q}+\int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}\left(f^{(n)},t\right)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}\right)^{\frac {1}{q}}$

The Besov spaces $B_{2,2}^{s}(\mathbf {R} )$ coincide with the more classical Sobolev spaces $H^{s}(\mathbf {R} )$.

If $p=q$ and $s$ is not an integer, then $B_{p,p}^{s}(\mathbf {R} )={\bar {W}}^{s,p}(\mathbf {R} )$, where ${\bar {W}}^{s,p}(\mathbf {R} )$ denotes the Sobolev–Slobodeckij space.

