Beppo Levi space - Alchetron, The Free Social Encyclopedia

In functional analysis, a branch of mathematics, a Beppo-Levi space, named after Beppo Levi, is a certain space of generalized functions.

In the following, D′ is the space of distributions, S′ is the space of tempered distributions in Rⁿ, D^α the differentiation operator with α a multi-index, and v ^ is the Fourier transform of v.

The Beppo-Levi space is

W ˙ r , p = { v ∈ D ′ : | v | r , p , Ω < ∞ } ,

where |⋅|_r,p denotes the Sobolev semi-norm.

An alternative definition is as follows: let m ∈ N, s ∈ R such that

− m + n 2 < s < n 2

and define:

H s = { v ∈ S ′ : v ^ ∈ L loc 1 ( R n ) , ∫ R n | ξ | 2 s | v ^ ( ξ ) | 2 d ξ < ∞ } X m , s = { v ∈ D ′ : ∀ α ∈ N n , | α | = m , D α v ∈ H s }

Then X^m,s is the Beppo-Levi space.

References

Beppo-Levi space Wikipedia

(Text) CC BY-SA