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, Dα the differentiation operator with α a multi-index, and v ^ {\displaystyle {\widehat {v}}} is the Fourier transform of v. The Beppo Levi space is
W ˙ r , p = { v ∈ D ′ : D α v ∈ L p for all | α | = r } . {\displaystyle {\dot {W}}^{r,p}=\left\{v\in D'\ :\ D^{\alpha }v\in L^{p}{\text{ for all }}|\alpha |=r\right\}.}
An alternative definition is as follows: let m ∈ N, s ∈ R such that
− m + n 2 < s < n 2 {\displaystyle -m+{\tfrac {n}{2}}<s<{\tfrac {n}{2}}}
and define:
H s = { v ∈ S ′ : v ^ ∈ L loc 1 ( R n ) , ∫ R n | ξ | 2 s | v ^ ( ξ ) | 2 d ξ < ∞ } X m , s = { v ∈ D ′ : ∀ α ∈ N n , | α | = m , D α v ∈ H s } {\displaystyle {\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 Xm,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