Beppo-Levi space

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 ${\displaystyle {\widehat {v}}}$ is the Fourier transform of v.

The Beppo Levi space is

${\displaystyle {\dot {W}}^{r,p}=\left\{v\in D'\ :\ |v|_{r,p,\Omega }<\infty \right\},}$

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

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

${\displaystyle -m+{\tfrac {n}{2}}<s<{\tfrac {n}{2}}}$

and define:

${\displaystyle {\begin{aligned}H^{s}&=\left\{v\in S'\ :\ {\widehat {v}}\in L_{\text{loc}}^{1}(\mathbf {R} ^{n}),\int _{\mathbf {R} ^{n}}|\xi |^{2s}|{\widehat {v}}(\xi )|^{2}\,d\xi <\infty \right\}\\[6pt]X^{m,s}&=\left\{v\in D'\ :\ \forall \alpha \in \mathbf {N} ^{n},|\alpha |=m,D^{\alpha }v\in H^{s}\right\}\\\end{aligned}}}$

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