We study the structure of the singular sets of solutions to elliptic differential equations in $\mathbb R^n$, the subsets where the solutions, with their gradients, vanish. We prove that such sets are countably $n-2$-rectifiable under the as sumption that the solutions vanish at finite order. We construct the tangent spaces for all points in the singular sets.

We present simplified proofs of the weighted-norm inequalities of R. Hunt, B.Muckenhoupt and R. Wheeden, Concerning singular integrals and Maximal functions. The inequalities in question are $$ \int_{\mathbb R^n}\left\lvert Tf(x)\right\rvert^p\omega(x)dx=C\int_{\mathbb R^n}\left\lvert f(x)\right\rvert^p \omega(x)dx, $$ where $T$ denotes eigher a singular integral operator, or the maximal function of Hardy and Littlewood, and $\omega$ satisfies appropriate (necessary and sufficient) conditions.