I am studying convolution of distributions to solve linear PDEs like the Laplace and Heat equations. A few justifications are skipped in my lecture notes and I'm trying to fill in the gaps:

1. u is homogeneous of degree \lambda ---> {{\partial}^\alpha}u is homog. of degree \lambda - |\alpha|.

2. {{\partial}^\alpha} \delta is homog. of degree -n-|\alpha|

For 1. I know what the definition of a homogeneous function (of degree \lambda) is: f(ax) = a^{\lambda}f(x). Also a distribution u, is homog. (of degree \lambda) if

<u, \phi_a> = 1/(a^n + \lambda) <u, \phi>. I can show that these definitions coincide when u is a function and am trying to use the same method for proving 1. but it doesn't quite work.... For 2. it's easy to show that the delta distribution is homog. of order -n, but that's not what I need to show.....

3. Show |x|^2 \Laplacian \delta = 2n\delta (in \mathbb{R^n}) . I don't have a clue for this one.....Thanks

p.s. I am almost computer illiterate, but does it take this long to fix \Latex.