Besov space

Template:Short description 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. This definition is quite limited because it does not extend to the range s ≤ 0.

Let

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

and define the modulus of continuity by

${\omega }_p^2(f,t)=\underset{|h|\le t}{\sup }{\Vert {\mathrm{\Delta }}_h^{2}f\Vert }_p$

Let Template:Mvar 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 Template:Mvar such that

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

Norm

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

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

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 $B_{p,p}^s(\mathbf{𝐑})={\overline{W}}^{s,p}(\mathbf{𝐑})$, where ${\overline{W}}^{s,p}(\mathbf{𝐑})$ denotes the Sobolev–Slobodeckij space.

References

• 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. Template:ISBN