Besov space

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In mathematics, the Besov space (named after Oleg Vladimirovich Besov) 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.


and define the modulus of continuity by

Let n be a non-negative integer and define: s = n + α with 0 < α ≤ 1. The Besov space contains all functions f such that


The Besov space is equipped with the norm

The Besov spaces coincide with the more classical Sobolev spaces .

If and is not an integer, then , where denotes the Sobolev–Slobodeckij space.


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  • 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