Of course, distributions are ok since I can regularize them

@Guido: you are right, but such objects are not functions, rather homogeneous distributions, since they are not locally integrable. In dim 1 you get vp(1/x), the principal value. The example I mentioned is essentially the next best thing: think of $\frac{x}{x^2+\epsilon^2}$

@Alexander: I am using the standard extension by Schwartz to tempered distributions

Let me add that functions satisfying self-improvement properties like the one considered here are rather important in harmonic analysis (I am thinking of reverse Hölder classes)

The first case works also if $(1+|y|)^\beta$ is replaced by $|y|^\beta$ for $\beta<n$

Of course, the $L^{q,2}$ norm is controlled by the $L^2$ norm of $\partial u$. This is a Sobolev embedding with values in Lorentz spaces