Certain classes of functional \({\mathcal B}\)-spaces (Q789674)

The author defines and investigates Banach spaces which generalize the Hölder spaces. Let X be a separable Banach space with the norm \(\|.\|\). By \(\Phi\) we denote the class of non-negative functions \(\phi\) definded on \((0,1>\times X\) which have the following properties: 1) \(\phi\) is continuous on (0,1), absolutely homogeneous in x and subadditive in both variables; 2) the mapping \(\phi\) (\(\tau\),.):\(X\to R\) is pointwise continuous on (0,1) and locally uniformly bounded on \((0,1>;\) 3) there exists \(x\in X\) such that \(\sup \{\phi(\tau,x);\tau \in(0,1>\}<\infty\) and \(\lim_{\tau \to 0}(\tau,x)\neq 0;\) 4) there exists a function \(\psi\) non-decreasing on \((0,1>\), \(\lim_{\tau \to 0}\psi(\tau)=0\), and such that the function \(\phi\) (\(\tau\),x)\(\psi\) (\(\tau)\) has all properties of the modulus of continuity. For \(\phi\in \Phi\) the classes \(X_{\phi}=\{x\in X;\quad \sup \{\phi(\tau,x);\quad \tau \in(0,1>\}<\infty \}\) and \(X^ 0\!_{\phi}=\{x\in X_{\phi};\lim_{\tau \to 0}\phi(\tau,x)=0\}\) with the norm \(\| x\|_{\phi}= \| x\| +\sup \{\phi(\tau,x);\tau \in(0,1>\}\) are Banach spaces. A sequence \(\{x_ n\}\subset X_{\phi}\) is said to be weak-(*) convergent to \(x_ 0\in X_{\phi}\) if \(x_ n\to x_ 0\) in X and \(\{x_ n\}\) is bounded in \(X_{\phi}.\) The author investigates the strong and weak-(*) convergence in \(X_{\phi}\), the dual spaces and the boundedness and compactness of operators in \(X_{\phi}\), \(X^ 0\!_{\phi}\). The results are then extended to the case of composition \(\phi {\mathbb{O}}\phi_ 1\) of \(\phi,\phi_ 1\in \Phi\), but it is not quite clear, how this composition is defined. An application is given for \(\phi(\delta,x)=\omega_ k(\delta,x)/\psi(\delta),\) where \(\omega_ k(\delta,x)=\sup \{\|(U_{\tau}-E)^ kx\|;\quad \tau \leq \delta \}, \{U_{\tau}\}\) is a strongly continuous semigroup of operators in X and E is the identity operator.