Sobolev spaces vs. Hardy Spaces I have seen sobolev spaces (the ones with the p norms of the derivatives of a multivariable function) and Hardy spaces (the objects investigated in harmonic analysis when one asks about tangential and radial limits of harmonic functions) both denoted by the same notation.  Is that because they are related?

No, the notation is not supposed to imply a relation.
Hardy spaces are denoted $H^p$ in honour of G.H. Hardy, who first studied them.
The Sobolev spaces $H^k$ of all $k$ times weakly differentiable $L^2$ functions with their weak derivatives up to order $k$ all being in $L^2$ is called $H^k$ because it is a Hilbert space.
This sets them apart from the Sobolev spaces $W^{k,p}$ of $k$-times weakly differentiable functions with all derivatives in $L^p$, which are not Hilbert spaces unless $p=2$.
This might be worth reading.
