Besov space

In mathematics, the Besov space (named after Oleg Vladimirovich Besov) B p , q s ( 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

Δ 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

ω p 2 ( f , t ) = sup | h | ≤ t ‖ Δ h 2 f ‖ 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 B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbf {R} )} contains all functions f such that

f ∈ W n , p ( R ) , ∫ 0 ∞ | ω p 2 ( f ( n ) , t ) t α | q d t t < ∞ . {\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 B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbf {R} )} is equipped with the norm

‖ f ‖ B p , q s ( R ) = ( ‖ f ‖ W n , p ( R ) q + ∫ 0 ∞ | ω p 2 ( f ( n ) , t ) t α | q d t t ) 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 B 2 , 2 s ( R ) {\displaystyle B_{2,2}^{s}(\mathbf {R} )} coincide with the more classical Sobolev spaces H s ( R ) {\displaystyle H^{s}(\mathbf {R} )} .

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

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