know that the blow up of $u$ centered at $x_i$ at the radius $d_{ij}\equiv d(x_i,x_j)$ looks close to $P^d$. In particular, since both $x_i$ and $x_j$ are ’good’ points relative to the frequency pinching, by construction, we have that $x_i$ and $x_j$ must be close to the plane $V$ relative to $d_{ij}$. Making this precise is exactly the statement that $P^V$ restricted to $\{x_i\}$ is $(1+\epsilon)$-biLipschitz. In particular, the Vitali covering $B_{r_i}(x_i)$ induces a Vitali covering $\{V_{r_i/2}(P^V(x_i))\}$ of the $n-2$-ball $B_1\cap V$. Thus we get from this the estimate (3).

Harmonic functions {#s:harmonic}
==================

In this Section, we concentrate on harmonic functions in ${\mathbb{R}}^n$, and will first prove Theorem \[t:main\_critical\] in this simplified case. This will allow us to illustrate the main ideas of the proof without the confusion of the added technical complications needed for the general case. More than that, many of the tools we will need for general solutions of (\[eq\_Lu\]) will follow by appropriate approximation arguments with harmonic functions, and thus many of the results of this Section are directly relevant.

We start in section \[ss:hhp\] by recalling some basic properties of homogeneous harmonic polynomials, hhPs in short. For a more complete overview on the subject, 