next subsections will refer to the generalized frequency.

Tangent Maps and Blow Ups {#ss:tangent_maps}
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In this subsection we define the notion of a blow up and discuss both new and old results related to it. To discuss this with precision, let us define for $x\in B_1(0)$ the linear transformation $$\begin{gathered}
\label{eq_Q}
 Q_{x} (y) = q_{ij}(y-x)^i e^j\, ,\end{gathered}$$ where $q^{ij}(x)$ is the square root of the matrix $a^{ij}(x)$. For instance if we consider just the Laplacian then $Q\equiv I$ is just the identity map. It is evident that $Q_x$ is a bi-Lipschitz equivalence from ${\mathbb{R}}^n$ to itself with Lipschitz constant $\leq (1+\lambda)^{1/2}$. Thus if $u$ solves (\[eq\_Lu\]) with $x\in B_{1/2}(0)$ and $r<\frac{1}{2(1+\lambda)^{1/2}}$ then we can define the blow up by

\[d:tangent\]

1.  For $x\in B_{1/2}(0)$ and $r<\frac{1}{2(1+\lambda)^{1/2}}$ we define $T^u_{x,r}:B_{r^{-1}}(0)\to{\mathds{R}}$ by $$\begin{gathered}
     T_{x,r}^u(y) = \frac{u(x+rQ_x(y))-u(x)}{{\left(\fint_{\partial B_1(0)} (u(x+rQ_x (y))-u(x))^2 \right)}^{1/2}}\, .\end{gathered}$$

2.  For $x\in B_1(0)$ we define $$\begin{gathered}
     T_x^u(y)=\lim_{r\to 0} T_{x,r}u(y)\, .\end{gathered}$$

By unique continuation and the maximum principle, $T^u_{x,r}$ is defined for all positive $r$ sufficiently small. The existence of the limit is a different matter. If the coefficients of the equation are smooth, its existence is an easy consequence of Taylor’s theorem and the unique continuation principle. In this 