Fourier transform of a tempered measure on $\R^n$. Recall that a (non-negative, not necessarily finite) measure $\mu$ is called tempered if $$\int_{\R^n} (1+|x|_2)^{-\beta}\ d\mu(x)< \infty$$ for some $\beta >0.$

Our definition of a star body $K$ assumes that the origin is an interior point of $K.$ If $0<p<n,$ then $\|\cdot\|_K^{-p}$ is a locally integrable function on $\R^n$ and represents an even homogeneous of degree $-p$ distribution. If $\|\cdot\|_K^{-p}$ represents a positive definite distribution for some $p\in (0,n),$ then its Fourier transform is a tempered measure which is at the same time a homogeneous distribution of degree $-n+p.$ One can express such a measure in polar coordinates, as follows.

\[posdef\] ([@K1 Corollary 2.26]) Let $K$ be an origin symmetric convex body in $\R^n$ and $p\in (0,n).$ The function $\|\cdot\|_K^{-p}$ represents a positive definite distribution on $\R^n$ if and only if there exists a finite Borel measure $\mu$ on $S^{n-1}$ so that for every even test function $\phi,$ $$\int_{\R^n} \|x\|_K^{-p} \phi(x)\ dx = \int_{S^{n-1}} \left(
\int_0^\infty t^{p-1} \hat\phi(t\xi) dt \right) d\mu(\xi).$$

For any even continuous function $f$ on the sphere $S^{n-1}$ and any non-zero number $p\in \R,$ we denote by $f\cdot r^{p}$ the extension of $f$ to an even homogeneous function of 