In mathematics, the Besov space (named after Oleg Vladimirovich Besov) ${\displaystyle B_{p,q}^{s}(\mathbf {R} )}$ 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

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

and define the modulus of continuity by

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

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

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

## Norm

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

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

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

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

## References

• Triebel, Hans (1992). Theory of Function Spaces II. doi:10.1007/978-3-0346-0419-2. ISBN 978-3-0346-0418-5.
• Besov, O. V. (1959). "On some families of functional spaces. Imbedding and extension theorems". Dokl. Akad. Nauk SSSR (in Russian). 126: 1163–1165. MR 0107165.
• DeVore, R. and Lorentz, G. "Constructive Approximation", 1993.
• DeVore, R., Kyriazis, G. and Wang, P. "Multiscale characterizations of Besov spaces on bounded domains", Journal of Approximation Theory 93, 273-292 (1998).
• Leoni, Giovanni (2017). A First Course in Sobolev Spaces: Second Edition. Graduate Studies in Mathematics. 181. American Mathematical Society. pp. 734. ISBN 978-1-4704-2921-8