Besov space

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


 * $$ \Delta_h f(x) = f(x-h) - f(x)$$

and define the modulus of continuity by


 * $$ \omega^2_p(f,t) = \sup_{|h| \le t} \left \| \Delta^2_h f \right \|_p$$

Let $n$ be a non-negative integer and define: $s = n + α$ with $0 < α ≤ 1$. The Besov space $$B^s_{p,q}(\mathbf{R})$$ contains all functions $f$ such that
 * $$ f \in W^{n, p}(\mathbf{R}), \qquad \int_0^\infty \left|\frac{ \omega^2_p \left ( f^{(n)},t \right ) } {t^{\alpha} }\right|^q \frac{dt}{t} < \infty.$$

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


 * $$ \left \|f \right \|_{B^s_{p,q}(\mathbf{R})} = \left( \|f\|_{W^{n, p} (\mathbf{R})}^q + \int_0^\infty \left|\frac{ \omega^2_p \left ( f^{(n)}, t \right ) } {t^{\alpha} }\right|^q \frac{dt}{t} \right)^{\frac{1}{q}}$$

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

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