A new interpolation approach to spaces of Triebel-Lizorkin type (Q5965355)

This paper is the continuation of [the author and \textit{A. Ullmann}, J. Fourier Anal. Appl. 20, No. 1, 135--185 (2014; Zbl 1316.46036)]. The concrete starting point are again the nowadays well-known homogeneous spaces \(\dot{F}^s_{p,q} (\mathbb R^n)\) normed by NEWLINE\[NEWLINE \Big\| \Big( \int^\infty_0 \big| t^{-s/2} \phi (t A) \big|^q \frac{dt}{t} \Big)^{1/q} \, | L_p (\mathbb R^n) \Big\| NEWLINE\]NEWLINE \(1<p<\infty\), \(1\leq q \leq \infty\), where \(A = -\Delta\) is the Laplacian and the homomorphic function \(\phi\) refers to the Gauss-Weierstrass semi-group and its derivatives, for example, \(\phi (z) = z e^{-z}\). The paper deals with the so-called \(\ell_q\)-interpolation in an abstract setting where \(L_p (\mathbb R^n)\) is replaced by a Banach function space \(X\) and \(A\) stands for so-called \({\mathcal R}_s\)-sectorial operators in \(X\) admitting a sufficiently powerful functional calculus. The \(\ell_q\)-interpolation method applies to the quasi-linearizable interpolation couple \(\big( X, D(A) \big)\) resulting in the indicated concrete situation to NEWLINE\[NEWLINE \dot{F}^{2\theta}_{p,q} (\mathbb R^n) = \big( L_p (\mathbb R^n), \dot{W}^2_p (\mathbb R^n) \big)_{\theta, \ell_q}, NEWLINE\]NEWLINE \(1<p<\infty\), \(1\leq q \leq \infty\), \(0<\theta<1\).