Composition operators on Hardy spaces on Lavrentiev domains

For any simply connected domain , we prove that a Little- wood type inequality is necessary for boundedness of composition operators on Hp( ), 1 p < 1, whenever the symbols are finitely-valent. Moreover, the corresponding ¿little-oh¿ condition is also necessary for the compactness. Nevertheless, it is shown that such inequality is not sufficient for characterizing bounded composition operators even induced by univalent symbols. Further- more, such inequality is no longer necessary if we drop the extra assumption on the symbol of being finitely-valent. In particular, this solves a question posed by Shapiro and Smith [12]. Finally, we show a striking link between the geometry of the underlying domain and the symbol inducing the composi- tion operator in Hp( ), and in this sense, we relate both facts characterizing bounded and compact composition operators whenever is a Lavrentiev do- main