Besov space

Inmathematics,theBesov space(named afterOleg Vladimirovich Besov)$B_{p,q}^{s}(\mathbf {R} )$is acompletequasinormedspace which is aBanach spacewhen1 ≤p,q≤ ∞.These spaces, as well as the similarly definedTriebel–Lizorkin spaces,serve to generalize more elementaryfunction spacessuch asSobolev spacesand 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 themodulus of continuityby

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

Letnbe a non-negative integer and define:s=n+αwith0 <α≤ 1.The Besov space$B_{p,q}^{s}(\mathbf {R} )$contains all functionsfsuch 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 classicalSobolev 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 theSobolev–Slobodeckij space.

References

• Triebel, Hans (1992).Theory of Function Spaces II.doi:10.1007/978-3-0346-0419-2.ISBN978-3-0346-0418-5.
• Besov, O. V. (1959). "On some families of functional spaces. Imbedding and extension theorems".Dokl. Akad. Nauk SSSR(in Russian).126:1163–1165.MR0107165.
• 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.ISBN978-1-4704-2921-8