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 $R^{n}$, $D^{α}$ the differentiation operator with $α$ a multi-index, and $$\widehat{v}$$ is the Fourier transform of $v$.

The Beppo Levi space is


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

where $|⋅|_{r,p}$ denotes the Sobolev semi-norm.

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


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

and define:


 * $$\begin{align}

H^s &= \left \{ v \in S' \ : \ \widehat{v} \in L^1_\text{loc}(\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{align}$$

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