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 $\hat{v}$ is the Fourier transform of v.

The Beppo Levi space is

${\dot{W}}^{r,p}=\{v\in D^{\prime }:D^{\alpha }v\in L^p\text{ for all }|\alpha |=r\}.$

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

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

and define:

$\begin{array}{cc}{H}^s& =\{v\in S^{\prime }:\hat{v}\in L_{\text{loc}}^1({𝐑}^n),\underset{{𝐑}^n}{\int }|\xi {|}^{2s}|\hat{v}(\xi ){|}^{2}d\xi <\mathrm{\infty }\}\\ [6pt]X^{m,s}& =\{v\in D^{\prime }:\mathrm{\forall }\alpha \in {𝐍}^n,|\alpha |=m,D^{\alpha }v\in H^s\}\\ \end{array}$

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

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

External links

• L. Brasco, D. Gómez-Castro, J.L. Vázquez, Characterisation of homogeneous fractional Sobolev spaces https://link.springer.com/content/pdf/10.1007/s00526-021-01934-6.pdf
• J. Deny, J.L. Lions, Les espaces du type de Beppo-Levy https://aif.centre-mersenne.org/item/10.5802/aif.55.pdf
• R. Adams, J. Fournier, Sobolev Spaces (2003), Academic press -- Theorem 4.31