It was shown in the book "Finite Elements Methods for Navier-Stokes Equations" by Girault and Raviart that
$$H_0^1(\Omega)=H_0(\operatorname{div};\Omega)\cap H_0(\operatorname{curl};\Omega).$$
The proof has used the Fourier transform method, which says that the Fourier transform's norm of each partial derivative is bounded by the sum of norms of divergence and curl itself, and this concludes that the vector is in $H_0^1(\Omega)$, see p.35-36 in the book.
But I don't get the idea from the proof, since it was only shown that the sums or differences, which are related to the divergence and curl, are in $L_2$, it doesn't mean that each component should be also like that. A combination of such sums and differences may be possible, but I can't see that. Also, there are no descriptions about the direct boundary values of the function but only values involved with $H_0(\operatorname{div};\Omega)$ and $H_0(\operatorname{curl};\Omega)$, but those boundary conditions are not really equivalent to boundary conditions in $H_0^1(\Omega)$.