Distributional control for operators on vector-valued \(L^p\)-spaces
Let \((\Omega,{\mathcal J},\mu)\) be a \(\sigma\)-finite measure space, and let \((X,\|\cdot\|)\) be a Banach space. The space \(L^p(\Omega,\mu,X)\) consisting of \(X\)-valued \(p\)th power integrable functions on \(\Omega\) is denoted for convenience by \(E^p\). The distribution \(\phi(f;\cdot)\) of a function \(f:\Omega\to X\) is defined by \(\phi(f; y)= \phi(N(f);y)= \mu(\{w\in \Omega:\| f(w)\|> y\})\), \(y>0\). An operator \(T: E^1\cap E^\infty\to E^1\cap E^\infty\) is said to be `\(\mu\)-distributionally controlled' if there are positive constants \(c\), \(\gamma\), such that, for \(y>0\), \(f\), \(g\) in \(E^1\cap E^\infty\), (i) \(\phi(T(f);y)\leq c\phi(f;\gamma y)\) and (ii) \(\phi(\min\{\| T(f)\|, \| T(g)\|\}; y)\leq c\phi(\min\{\| f\|, \| g\|\};\gamma y)\). If \(G\) is a locally compact Abelian group, then a representation \(u\to R_u\) of \(G\) is said to be \(\mu\)-distributionally controlled if (i) and (ii) are valid for \(T= R_u\), \(u\in G\). In the first main theorem of this paper, the authors show that if \(T\) is \(\mu\)-distributionally controlled, with constants \(c\), \(\gamma\) as in (i), (ii), then for \(1\leq p<\infty\), there is a linear operator \(T^{(p)}\) on \(E^p\) such that \(T^{(p)}(f)= T(f)\) for \(f\in E^1\cap E^\infty\) and \(\| T^{(p)}\|\leq c^{1/p}\gamma^{- 1}\). In another theorem, it is shown that if \(u\to S_u\) is a representations of \(G\) on \(E^p\), then there are \(\mu\)-distributionally controlled representations of \(G\) such that \(S_u= R^{(p)}_u\), \(u\in G\), if and only if each operator \(S_u\) is separation preserving and bounded in \(E^p\), \(1\leq p<\infty\), and on \(E^p\cap E^\infty\) with \(L^\infty\)-norm. In a result relating to bounded representations, it is shown that if \(u\to R_u\) is a representation of \(G\) which is uniformly bounded on \(E^p\) for \(p= p_1\) or \(p= p_2\), \(1\leq p_1\), \(p_2<\infty\), then the representation is strongly continuous on \(E^{p_1}\) if and only if it is strongly continuous on \(E^{p_2}\). Scalar-valued function forms of the results of the paper are indicated as contained in a paper of \textit{N. Asmar}, \textit{E. Berkson} and \textit{T. A. Gillespie} [Integral Equations Oper. Theory 14, No. 3, 311-341 (1991; Zbl 0763.47038)].
operators on vector-valued \(L^p\)-spaces
\(\mu\)-distributionally controlled representation
scientific article; zbMATH DE number 1051525