Let $X$ be a Polish space, and $\mu$ a locally finite measure. Take any $p \in \{0\} \cup [1,\infty)$. We will say that a linear operator $T \colon L^p(\mu) \to L^p(\mu)$ has property $(\ast)$ if every $x \in X$ admits a neighbourhood $U$ such that for every $f \in L^\infty(\mu)$ we have $T(f.\mathbf{1}_U) \in L^\infty(\mu)$.
(If $T$ maps real-valued functions to real-valued functions in a way that respects their partial ordering, then it is sufficient that every $x \in X$ has a neighbourhood $U$ for which $T(\mathbf{1}_U) \in L^\infty(\mu)$.)
Is there an actual term for property $(\ast)$? And/or, is there a term for nonsingular transformations $(X,\mathcal{B}(X),\mu,f \colon X \to X)$ whose transfer operator $T_f \colon L^1(\mu) \to L^1(\mu)$ has property $(\ast)$?
