Besov space

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

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

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

and define the modulus of continuity by

${\displaystyle \omega _{p}^{2}(f,t)=\sup _{|h|\leq t}\left\|\Delta _{h}^{2}f\right\|_{p}}${\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 ${\displaystyle B_{p,q}^{s}(\mathbf {R} )}${\displaystyle B_{p,q}^{s}(\mathbf {R} )} contains all functions f such that

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

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

${\displaystyle \left\|f\right\|_{B_{p,q}^{s}(\mathbf {R} )}=\left(\|f\|_{W^{n,p}(\mathbf {R} )}^{q}+\int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}\left(f^{(n)},t\right)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}\right)^{\frac {1}{q}}}${\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 ${\displaystyle B_{2,2}^{s}(\mathbf {R} )}${\displaystyle B_{2,2}^{s}(\mathbf {R} )} coincide with the more classical Sobolev spaces ${\displaystyle H^{s}(\mathbf {R} )}${\displaystyle H^{s}(\mathbf {R} )}.

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

• 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

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