The author extends lemmas by J. Bourgain, H. Brezis amd P. Mironescu [in: Optimal control and partial differential equations, Proceedings of the conference, Paris, France, December 4, 2000, IOS Press, Amsterdam, 439–455 (2001; Zbl 1103.46310)] who proved that \[ \lim_{s\rightarrow1}(1-s)^{1/p}\| f\| _{W_{0}^{s,p} (\mathbb{R}^{n})}\simeq\| \nabla f\| _{L^{p}(\mathbb{R}^{n})} \] and V. G.Maz’ya and T. O.Shaposhnikova [J.Funct.Anal.195, No. 2, 230–238 (2002; Zbl 1028.46050)] who proved that \[ \lim_{s\rightarrow0}s^{1/p}\| f\| _{W_{0}^{s,p}(\mathbb{R}^{n} )}\simeq \| f\| _{L^{p}(\mathbb{R}^{n})} \] to the setting of interpolation scales. For the real interpolation method, the following is shown (a similar result is obtained for complex interpolation). Suppose that \(\vec{X}=(X_{0},X_{1})\) is a normal pair. Then
(1) For \(1\leq q<\infty,\) \(f\in X_{0}\cap X_{1}\), we have that \[ \lim_{s\rightarrow1}\| f\| _{\vec{X}_{\theta,q}}=\| f\| _{X_{1}}. \]
(2) For \(1\leq q<\infty\), \(f\in X_{0}\cap X_{1}\), we have that \[ \lim_{s\rightarrow0}\| f\| _{\vec{X}_{\theta,q}}=\| f\| _{X_{0}}. \]
(3) For \(1\leq q<\infty,\) \(f\in X_{0}\cap( \bigcup_{s\in(0,1)}{\vec X}_{s,q})\), we have that \[ \lim_{s\rightarrow0}\| f\| _{\vec{X}_{\theta,q}}=\| f\| _{X_{0}}. \]
The Bourgain–Brezis–Mironescu result [op.cit.] follows from this theorem using that \[ (1-s)^{1/p}\| f\| _{W_{0}^{s,p}(\mathbb{R}^{n})}\simeq (n+sp)^{1/p}p^{-1/p}(L^{p}(\mathbb{R}^{n}),W_{0}^{1,p}(\mathbb{R} ^{n})) _{s,p}. \]
Similarly, one also recovers the resul of V. G.Maz’ya and T. O.Shaposhnikovat [op.cit.], since \[ s^{1/p}\| f\| _{W_{0}^{s,p}(\mathbb{R}^{n})}\simeq p^{-1/p} n^{1/p} (L^{p}(\mathbb{R}^{n}),W_{0}^{1,p}(\mathbb{R}^{n})) _{s,p}. \]
Several connections with extrapolation theory and new applications to limits of Sobolev scales are also obtained.