Limits of higher-order Besov spaces and sharp reiteration theorems
The authors compute the limits of higher-order Besov norms and derive the sharp constants for certain forms of the Sobolev embedding theorem using interpolation methods. Their result extends, to higher-order spaces, the work by \textit{H. Brézis, J. Bourgain, P. Mironescu} in [Optimal control and partial differential equations. In honour of Professor Alain Bensoussan's 60th birthday. Proceedings of the conference, Paris, France, Amsterdam: IOS Press; Tokyo: Ohmsha, 439--455 (2001; Zbl 1103.46310)] that states that \[ \lim_{s\rightarrow 1^{-}}(1-s)^{1/p}\| f\| _{W^{s,p}}\simeq \| \nabla f\| _{L^{p}} \] and \textit{V. Maz'ya, T. Shaposhnikova} [J. Funct. Anal. 195, No.~2, 230--238 (2002; Zbl 1028.46050); erratum, J. Funct. Anal. 201, No. 1, 298--300 (2003)] that states that \[ \lim_{s\rightarrow 0^{+}}s^{1/p}\| f\| _{W^{s,p}}\simeq \| f\|_{L^{p}}. \] The main results says: If \(k\in \mathbb{N},\) \(s\in (0,1),\) and \(p,q\in [ 1,\infty ),\) then \[ \lim_{s\rightarrow 0}(s(1-s))^{1/q}\| f\| _{B_{p,q}^{sk}}\simeq \| f\| _{L^{p}},\text{ \;}f\in C_{0}^{\infty }(\mathbb{R}^{n}), \] and \[ \lim_{s\rightarrow 1}(s(1-s))^{1/q}\| f\| _{B_{p,q}^{sk}}\simeq \| f\| _{W^{k,p}},\text{ \;}f\in C_{0}^{\infty }(\mathbb{R}^{n}). \] Let \(k\geq 1,s\rightarrow 1,\) \(1\leq p<\infty ,\) \(kp<n\) and \(q>0.\) Then \[ B_{p,q}^{sk}\subset (1-s)^{-\alpha }L^{r_{s},q},\text{ \;}1/r_{s}=1/p-sk/n. \] Let \(k\geq 2,\) \(s\rightarrow 1,\) \(kp=n,\) \(p>1,\) \(1/r_{s}=1/p-sk/n\) and \(q>0.\) Then \[ B_{p,q}^{sk}\subset (1-s)^{-\alpha +1}L^{r_{s},q},\text{ \;}\alpha =\min \{1/p,1/q\}. \] Let \(s\rightarrow 1,\) \(1/r_{s}=1-s,\) \(k=n.\) Then \(B_{1,q}^{sn}\subset L^{r_{s},q}.\) Let \(k\geq 1,sk\rightarrow n/p,\) \(kp>n,\) \(1/r_{s}=1/p-sk/n\) and \(q>0.\) Then \[ B_{p,q}^{sk}\subset r_{s}^{-c}L^{r_{s},q},\text{ \;}c=\max \{1,1/q\}. \] Let \(k\geq 1,s\rightarrow 0,\) \(1\leq p<\infty ,\) \(sk<n/p,\) \(q>0.\) Then \[ B_{p,q}^{sk}\subset s^{-\alpha }L^{r_{s},q},\text{ \;}1/r_{s}=1/p-sk/n,\text{ }\alpha =\min \{1/p,1/q\}. \]
scientific article; zbMATH DE number 2172977