Operadores lineares em espaços de Hardy

Abstract

The present work aims to present an example of linear a functional defined on a dense subspace of the Hardy space H1(Rn) to be built, with the intention of showing that despite the fact that this functional is uniformly bounded on all atoms, it does not extend to a bounded functional on the whole H1(Rn). This example was published by Bownik, M.B [2]. Therefore, this shows that in general is not enough to verify that an operator or a functional is bounded on atoms to conclude that it extends boundedly to the whole space. The construction is based on the fact due to Y. Meyer [1] which states that quasi-norms corresponding to finite and infinite atomic decomposition in Hp(Rn), 0 < p 6 1 are not equivalent. On the other hand it will be given a necessary and suficient condition for when and operator T defined in a dense Hardy subspace Hp(Rn) for 0 < p 6 1 is bounded extended. Such conditions were published by D. Yang and Y. Zhou [3].