For tempered distributions on $\mathbb{R}^n$ we can write $\widehat{\nabla f}(p)=p\hat{f}(p)$ and hence by Plancherel, we have equations like $(\nabla f,\nabla g)=(p\hat{f}(p),p\hat{g}(p))$ for functions in $H^1$.
Do we have a similar tool for calculating $$\int_\Omega \overline{\nabla f(x)}\cdot\nabla g(x) dx$$ when $\Omega$ is a bounded domain in $\mathbb{R}^n$ f and g are $H^1$-functions on $\Omega$? I'd like to extend to $\mathbb{R^n}$ but this gives me a contribution at the boundary which I don't know how to handle.
thanks for your help