Are there any known inequalities of the following type for $f$ satisfying some conditions:

$$ \|f\|_{H_p(\mathbb{R})} \le C\|f\|_{L_p(\mathbb{R})}, $$

where $H_p$ denotes the real Hardy space and $p\le 1$ (otherwise the question would be trivial)?

Here is an example:

If $f$ is bandlimited, then

$$ \|f\|_{H_p(\mathbb{R})} \sim \|f\|_{L_p(\mathbb{R})}. $$

Does a similar conclusion also hold if $f$ is sufficiently smooth?

Of particular interest to me is the following case:

Assume that $\varphi$ satisfies the inequality above. Is the same true for any function in the integer-shift-invariant space spanned by the integer shifts of $\varphi$?