Question

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 $H^1$.

Answer 1

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".