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After some confusion on my part, I wanted to know is there a profound mathematical reason why both Hardy spaces and Sobolev spaces are denoted by $H^p$(1). Is it just coincidence? Does it have any historical meaning?

Note: Granted that a Sobolev space of $k$ derivative all in $L^p$ will not be denoted this way, but usually when $p=2$ then it is omitted, and we are left with $H^k$.

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    $\begingroup$ according to this earlier MO question, the letter $H$ is used for the $p=2$ Sobolev space only because it's a Hilbert space. $\endgroup$
    – Carlo Beenakker
    Commented Jan 16, 2017 at 7:28

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