View source for Beppo-Levi space

In [[Functional analysis|functional analysis]], a branch of mathematics, a '''Beppo Levi space''', named after Beppo Levi, is a certain space of [[Generalized function|generalized function]]s.
In the following, {{mvar|D′}} is the space of [[Distribution (mathematics)|distributions]], {{mvar|S′}} is the space of tempered distributions in {{math|'''R'''<sup>''n''</sup>}}, {{mvar|D<sup>α</sup>}} the differentiation operator with {{mvar|α}} a multi-index, and <math>\widehat{v}</math> is the [[Fourier transform]] of {{mvar|v}}.

The Beppo Levi space is

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

where {{math|{{!}}⋅{{!}}<sub>''r'',''p''</sub>}} denotes the [[Sobolev space|Sobolev]] semi-norm.

An alternative definition is as follows: let {{math|''m'' ∈ '''N''', ''s'' ∈ '''R'''}} such that

: <math>-m + \tfrac{n}{2} < s < \tfrac{n}{2}</math>

and define:

:<math>\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}</math>

Then {{math|''X''<sup>''m'',''s''</sup>}} 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


[[Category:Functional analysis]]


{{Sourceattribution|Beppo-Levi space}}