An algebra of absolutely continuous functions and its multipliers (Q812161)

Let \(I=[0,1]\) with the usual topology and let \(C(I)\) be the set of all continuous complex-valued functions on \(I\). For \(1 \leq p \leq \infty\), let \(AC_p = \{ f \in C(I): f\) is absolutely continuous on \(I\), \(f(0) =0\) and \(f' \in L^p(I) \}\). A norm can be defined on \(AC_p\) by \(| | | f| | | = \|f' \|_p\). The author proves that \(AC_p\) is a Banach subalgebra of \(C(I)\) and that the maximal ideal space of \(AC_p, 1 \leq p \leq \infty\), is homeomorphic to \((0, 1]\). The author also shows that \(AC_r \subseteq AC_p\) for \(r >p\) and that \(AC_p\) has an approximate identity for \(1 \leq p < \infty\). There is no approximate identity for \(AC_{\infty}\). Since \(AC_p\) is semisimple, a mapping \(T: AC_p \rightarrow AC_p\) is a multiplier if \(Tg = mg\), \(g \in AC_p\), for some continuous bounded function \(m\) on \((0,1]\). The author characterizes multipliers \(T: AC_p \rightarrow AC_p\), \(1\leq p \leq \infty\), by proving: \(T\) is a multiplier if and only if there exists \(m \in C_b (0,1]\) such that for each \(\varepsilon > 0, m\) is absolutely continuous on \([\varepsilon,1], m' \in L^p[\varepsilon,1]\) and \(\|m'\chi_{[\varepsilon,1]} \|_p = O(e^{-{1\over p'}})\). (Here \(p'\) is the conjugate index of \(p\). Treat \(\varepsilon^{-{1\over p'}} = 1\) for \(p=1\).) Multipliers from \(AC_r\) to \(AC_p, r \neq p\), are given by continuous functions \(m \in C(0,1]\) which may not be bounded. In this paper it is shown that if \(T : AC_r \rightarrow AC_p\) is a multiplier, \(r < p \leq \infty\), then \(T=0\). It is also proven that if \(m \in C(0, 1]\) satisfies certain technical conditions, then \(m\) can be used to define a multiplier from \(AC_r\) to \(AC_p\) when \(r > p > 1\).