Limiting case for interpolation spaces generated by holomorphic semigroups
Let \(A\) be the generator of an analytic semigroup \(\{T(t)\); \(t\geq 0\}\) of bounded linear operators on a Banach space \((X, \|\cdot\|)\). Denote by \(X_{\theta, p}\) \((0 < \theta< 1)\) the subspace of all elements \(x\in X\) satisfying \[ \|x\|_{\theta,p} := \left(\int^{\infty}_0 (t^{1-\theta} \|AT(t) x\|)^p t^{-1} dt\right)^{1/p}< \infty, \quad p < \infty \] and \[ \|x\|_{\theta,\infty} := \sup_t t^{1-\theta} \|AT(t)x\|. \] The author investigates the spaces generated by the norm \(\|x\|_{\theta, p}\) in the limiting case \(\theta = 0\). In particular, it is proved that these spaces are spaces of maximal regularity for the semigroups \(T_p(t)\); for different operators \(A\) and \(B\) sufficient conditions for the equality \(X^{A}_{0, p} = X^{B}_{0, p}\) are given. This paper continues the investigation begun in [ibid. 47, No. 1, 105-114 (1993; Zbl 0816.47044)].
scientific article; zbMATH DE number 1268033