Homogeneity of distributions

We now wish to pass from the symbol $p$ of a classical $\Psi DO$, with a given symbol expansion

$$p(x,\xi) = \sum_{j=0}^{N-1} p_{d-j}(x,\xi) + r_N(x,\xi), \quad r_N \in S^d(U).$$

to the operator kernel (), by taking an inverse Fourier transform. However, the terms in this expansion may give divergent integrals when $y = x$. Therefore, we first need to look more closely at the inverse Fourier transforms of negative powers of $\vert\xi\vert$.

Assume that $n \geq 2$, for the rest of this section.

We can extend this definition to (tempered) distributions on $\bR^n$. Write $\phi_t$ for the dilation of $\phi$ by the scale factor $t$, that is, $\phi_t(\xi) := \phi(t\xi)$, so that the $\la$-homogeneity condition can be written as $\phi_t = t^\la \phi$ for $t > 0$.

The change-of variables formula for functions,

$$\int_{\bR^n} u(t\xi) \phi(\xi) d^n\xi = \int_{\bR^n} t^{-n} u(\eta) \phi(\eta/t) d^n\eta,$$

suggests the following definition of homogeneity.

Suppose now that $u$ is a smooth function on $\bR^n \setminus \{0\}$, such that

$$u(\xi) = r^\la v(\om), \words{for} \xi = r\om, r = \vert\xi\vert > 0, \om \in \bS^{n-1}.$$

We would like to extend it to a (tempered) distribution on the whole $\bR^n$. There are several cases to consider.