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### Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is
\begin{equation}
\omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})}
\end{equation}
where ...

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### Continuous embedding of Hardy space in Lebesgue space

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 ...

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### Connected components of space of maps between two manifolds

Question: What are the connected components of the familiar spaces of functions between two (let's say compact and smooth, for simplicity) manifolds $M$ and $N$?
Specifically, I'm thinking of the ...