H ^p (Rⁿ) is equidistributed withL ^p (Rⁿ)

Abstract

Let 0<p<∞. LetH ^p (R ⁿ) be the real variable Hardy spaces defined by Stein and Weiss. Let L^p(R ⁿ) be the usual Lebesgue space. It is shown that forf∈L ^p there is an$\tilde f \in H^p$ with the distribution functions of |f| and$\left| {\tilde f} \right|$ identical and$\left\| {\tilde f} \right\|_{H^p } \approx \left\| {\tilde f} \right\|_{L^p }$. The converse is trivially true.