On the Operator L(f)=f log |f|

The present note deals with the operator L: H1(Rn)→D′(Rn) which takes a given element f of the Hardy space to the function f log |f|. In general, this function need not be locally integrable. Nevertheless, due to peculiar cancellations of large positive and negative terms in the integral ∫ ϕf log |f| with ϕ∈C∞0(Rn), we are able to give meaning to f log |f| as a Schwartz distribution. We find several alternatives for this interpretation of f log |f|.