Both, Hardy's inequality and Sobolev's inequality, are estimates that compare the Laplacian of a function and to the function itself, admittedly in a slightly different fashion.
Still they seem to be used in similar contexts (e.g. Schrödinger operators) and I was wondering, if there are certain classes of problems where one would favor one over the other. What is their basic difference in mathematical content?
One possible formulation of the two inequalities is $(\forall \phi \in C_0^{\infty}(\mathbb{R}^d), d\geq 3)$:
$$\int_{\mathbb{R}^d}\vert \nabla \phi \vert^2 dx\geq \left(\frac{d-2}{2}\right)^2 \int_{\mathbb{R}^d} \frac{\vert \phi(x)\vert^2}{\vert x\vert^2}dx \qquad \text{(Hardy)}$$
$$C_d \int_{\mathbb{R}^d} \vert \nabla \phi \vert^2 \geq \left( \int_{\mathbb{R}^d} \vert \phi(x)\vert^{2d/(d-2)} \right)^{\frac{d-2}{d}} \qquad \text{(Sobolev)} $$
