Generalization of Sobolev spaces
In mathematics, the Besov space (named after Oleg Vladimirovich Besov)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9327f37c47f7fec0f03ec864cbb65b1036d8ebb7)
and define the modulus of continuity by
![{\displaystyle \omega _{p}^{2}(f,t)=\sup _{|h|\leq t}\left\|\Delta _{h}^{2}f\right\|_{p))](https://wikimedia.org/api/rest_v1/media/math/render/svg/a24c2e1befe5ed9139338ed9c36608457ddc625f)
Let n be a non-negative integer and define: s = n + α with 0 < α ≤ 1. The Besov space
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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed86c740608fb6e7f691bd863de3f56062f2d7)
Norm
The Besov space
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d222faf9223fbe761541ffcb48c9095ad5ca0b8c)
The Besov spaces
coincide with the more classical Sobolev spaces
.
If
and
is not an integer, then
, where
denotes the Sobolev–Slobodeckij space.