Interpolation in class of the Besov and Lizorkin-Triebel spaces. The limit case \(p_0= \infty\) (Q1603039)

From the text: In earlier papers [V. L. Krepkogorskiĭ, Russ. Acad. Sci., Sb., Math. 82, No. 2, 315--326 (1995; Zbl 0853.46030); translation from Mat. Sb. 185, No. 7, 36--76 (1994); Russ. Math. 43, No. 11, 38--46 (1999); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1999, No. 11, 41--49 (1999; Zbl 0995.46053)] we showed that by the interpolation of the couple of Besov spaces \((B^{s_0}_{p_0}(\mathbb R_n),B^{s_1}_{p_1}(\mathbb R_n))_{\theta,q}, 0<p_i<\infty, s_0\neq s_1, p_0\neq p_1\), we obtain spaces of the type \(BL^{s,k}_{p,q}\). Spaces of this type can also be obtained by interpolation of the Lizorkin-Triebel spaces \((F^{s_0}_{p_0,q_0},F^{s_1}_{p_1,q_1})_{\theta,q}, 0<p_i<\infty\). In this paper we extend these results to the case \(p_0=\infty\). A number of known spaces can be identified with \(F^s_{\infty,q}(\mathbb R_n)\) or \(B^s_{\infty}(\mathbb R_n)\), for example, the Hölder-Zygmund spaces \(G^s=B^s_{\infty}, \text{BMO}=F^s_{\infty,2}\) and \(L_p=F^0_{p,2}\). Here we obtain interpolation theorems for the pairs of spaces of the form \((G,F), (G,B),\;(\text{BMO},F)\) and \((G,L_p)\).