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

The Beppo-Levi space is $\dot{W}^{r,p}(\Omega) = \{v \in D'(\Omega) : |v|_{r,p,\Omega} < \infty \}$
($|.|_{r,p,\Omega}$ denotes the Sobolev semi-norm)

An alternative definition is:

Given $m \in N, s \in R$ such that

$-m + \frac{n}{2} < s < \frac{n}{2}$

Let

$H^s = \{ v \in S' | \hat{v} \in L^1_\text{loc}(R^n), \int_{R^n} |\xi|^{2s}| \hat{v} (\xi)|^2 \, d\xi < \infty \}$

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

$X^{m,s} = \{ v \in D' | \forall \alpha \in N^n, |\alpha| = m, D^{\alpha}v \in H^s \}$

## References

• Wendland, Holger (2005), Scattered Data Approximation, Cambridge University Press.
• Rémi Arcangéli; María Cruz López de Silanes; Juan José Torrens (2007), "An extension of a bound for functions in Sobolev spaces, with applications to (m,s)-spline interpolation and smoothing" Numerische Mathematik
• Rémi Arcangéli; María Cruz López de Silanes; Juan José Torrens (2009), "Estimates for functions in Sobolev spaces defined on unbounded domains" Journal of Approximation Theory