case, the limit is unique and, up to rescaling, $T_x^u$ is just the leading order polynomial of the Taylor expansion of $u-u(x)$ at $x$. In the general case, the existence of the limit has been proved in [@han_sing] and is a deeply important property of solutions to .

Using a simple change of variables, it is easy to see that the function $T$ satisfies an elliptic PDE of the form: $$\begin{gathered}
\label{eq_LT}
 \tilde {\mathcal{L}}(u)=\partial_i{\left(\tilde a^{ij} \partial _j T\right)} + \tilde b^i \partial_i T +\tilde c T =0\, ,\end{gathered}$$ with $\tilde a^{ij}(0)=\delta^{ij}$. Moreover, the conditions imply similar estimates for the coefficients $\tilde a^{ij}, \ \tilde b^i$: $$\begin{gathered}
\label{eq_aT}
 (1+\lambda r)^{-1}\delta^{ij}\leq \tilde a^{ij}\leq (1+\lambda r)\delta^{ij}, \, \text{Lip}(\tilde a^{ij})\leq \lambda r\, , \,
{\left|\tilde b^i\right|},{\left|\tilde c\right|} \leq \lambda r\, .\end{gathered}$$

An important property of the blow ups is that they are controlled by the frequency. We say the frequency at $x$ is $\delta$-pinched on the scales $[r_0,r_1]$ if $|N(x,r_0)-N(x,r_1)|<\delta$. It is known, see [@chnava] for instance, that for every $\epsilon>0$ there is a $\delta>0$ such that if the frequency is $\delta$-pinched at $r$, then $T^u_{x,r}$ is $\epsilon$ close to some homogeneous harmonic polynomial $P_d$. The primary weakness of this result from [@chnava], besides 