The present note deals with the operator ℒ: ℋ1(ℝn) → script D sign1(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 (Latin small letter esh) φ f 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|.
- Hardy space
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