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We obtain a new square function characterization of the weak Hardy space \(H^{p,\infty }\) for all \(p\in (0,\infty )\). This space consists of all tempered distributions whose smooth maximal function lies in weak \(L^p\). Our proof is based on interpolation between \(H^p\) spaces. The main difficulty we overcome is the lack of a good dense subspace of \(H^{p,\infty }\) which forces us to work with general \(H^{p,\infty }\) distributions.

Square Function Characterization of Weak Hardy Spaces