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I would like to have a reference to the following statement which I think is true: $$h^1 \hookrightarrow L^1.$$ The closest I came to this is in D. Goldberg's paper, "A local version of real Hardy spaces", where in Theorem 2 (p. 33) there is a characterization of the local hardy spaces $h^1$ from which in particular follows that $h^1$ is contained in $L^1$. However, a proof is not given (only some obscure reference to some supposedly similar proof in a well-known book of E. Stein) and I really wanted to know whether I can take for granted that one really has a continuous embedding from $h^1$ into $L^1$.

I really mean $h^1$, and not $H^1$.

share|cite|improve this question… notes that $H^{1}(\mathbb{R}^d) \subset L^{1}(\mathbb{R}^d)$ by the duality theorem of Fefferman and Stein. – Benjamin Dickman Sep 26 '12 at 8:52
Here: – Piero D'Ancona Sep 26 '12 at 15:58

Ok, I have found what I wanted. Just in case it can be useful to others, here it goes: it is clearly stated in Remark 2.5.8/4 (pp. 93-94) in Triebel's "Theory of Function Spaces".

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