Let $f \in C^{\gamma}_c(\mathbb{R}) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere except at $\vec{0}$
2) K homogeneous of degree $-n$, in particular $|K(x)| \leq \frac{c}{|x|^{n}}$
3) K has mean value zero on the unit sphere, ie $\int_{|x|=1}K(x)dS=0$
I was wondering if the Cauchy principal value of the convolution of $K$ with $f$ is "invariant" under a change of variables. That is, for a $C^1$ diffeomorphism $G: \mathbb{R}^n \rightarrow \mathbb{R}^n$, denoting $y=G(w)$ and $x=G(v)$, do we have:
\begin{eqnarray} \text{P.V.} \int_{\mathbb{R}^n} K(x-y)f(y)dy &\equiv& \lim_{\delta \searrow 0} \int_{|x-y|> \delta} K(x-y)f(y)dy \\ &=& \lim_{\delta \searrow 0} \int_{|v-w|> \delta} K \left(x-G(w) \right)f \left( G(w) \right) \left|\det \nabla G(w) \right| dw \quad \text{?} \end{eqnarray}