Complex interpolation on Besov-type and Triebel-Lizorkin-type spaces (Q2855475)

Let \(\theta\in(0,1)\), \(s_{0},s_{1}\in\mathbb{R}\), \(\tau_{0},\tau_{1}\in[0,\infty)\), \(p_{0},p_{1}\in(0,\infty)\), \(q_{0},q_{1}\in(0,\infty]\), \(s=s_{0}(1-\theta)+s_{1}\theta\), \(\tau=\tau_{0}(1-\theta)+\tau_{1}\theta\), \(1/p=(1-\theta)/p_{0}+\theta/p_{1}\) and \(1/q=(1-\theta)/q_{0}+\theta/q_{1}\). The authors, under the condition \(\tau_{0}p_{0}=\tau_{1}p_{1}\), identify the interpolated space by the complex method of a couple of Triebel-Lizorkin-type spaces \(F_{p,q}^{s,\tau}(\mathbb{R}^{n})\). They prove that NEWLINE\[NEWLINE\begin{multlined}\left[ \overset{\circ}{F}_{p_{0},q_{1}}^{s_{0},\tau_{0}}(\mathbb{R}^{n}),\overset{\circ}{F}_{p_{1},q_{1}}^{s_{1},\tau_{1}}(\mathbb{R}^{n})\right] _{\theta}=\overset{\circ}{F}_{p,q}^{s,\tau}(\mathbb{R}^{n})\\ =\left[ {F}_{p_{0},q_{1}}^{s_{0},\tau_{0}}(\mathbb{R}^{n}),\overset{\circ}{F}_{p_{1},q_{1}}^{s_{1},\tau_{1}}(\mathbb{R}^{n})\right] _{\theta}=\left[ \overset{\circ}{F}_{p_{0},q_{1}}^{s_{0},\tau_{0}}(\mathbb{R}^{n}),F_{p_{1},q_{1}}^{s_{1},\tau_{1}}(\mathbb{R}^{n})\right] _{\theta},\end{multlined}NEWLINE\]NEWLINE where \(\overset{\circ}{F}_{p,q}^{s,\tau}(\mathbb{R}^{n})\) denotes the closure of the Schwartz functions in \(F_{p,q}^{s,\tau}(\mathbb{R}^{n})\).