On the operator L (f) = f log vertical bar f vertical bar

The present note deals with the operator L : H-1(R-n) --> L'(R-n) which tak es a given element f of the Hardy space to the function f log \f\. In gener al; this function need not be locally integrable. Nevertheless, due to pecu liar cancellations of large positive and negative terms in the integral int egral phi f log \f\ with phi is an element of C-0(infinity)(R-n), we are ab le to give meaning to f log \f\ as a Schwartz distribution. We find several alternatives for this interpretation of f log \f\. (C) 1999 Academic Press .