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 $|\cdot| r, p$ denotes the Sobolev semi-norm.

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

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

and define:

{\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.

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

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