Polynomial growth functions define tempered distributions

For each integer $N\ge 0$, expansion of $(1+\mathrm{\mid }{x}_1\mathrm{\mid }+\cdots +\mathrm{\mid }{x}_n\mathrm{\mid }{)}^N$ gives a finite constant $A_{n,N}$ for which the following estimate holds.

The estimate in step 1.1 makes the integral absolutely convergent for every Schwartz test and proves that $u_f$ is tempered. It also shows directly that changing $f$ on a null set changes no pairing.

The integer shells $m\le \mathrm{\mid }x\mathrm{\mid }<m+1$ have measure at most $(2m+2{)}^n$. Hence $\int (1+\mathrm{\mid }x\mathrm{\mid }{)}^{-s}dx<\mathrm{\infty }$ whenever $s>n+1$, by comparison with ${\sum }_{m\ge 1}{m}^{n-s}$. If $\mathrm{\mid }f(x)\mathrm{\mid }\le C(1+\mathrm{\mid }x\mathrm{\mid }{)}^d$ off a compact set, choose an integer $N>d+n+1$. Local integrability handles the compact part, and the shell estimate handles its complement, so step 2.1 applies.

If $f\in L^1$, take $N=0$. If $1<p<\mathrm{\infty }$ and $q$ is conjugate to $p$, choose $N$ with $Nq>n+1$; Hölder gives $\int \mathrm{\mid }f\mathrm{\mid }(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-N}\le \parallel f{\parallel }_p\parallel (1+\mathrm{\mid }\cdot \mathrm{\mid }{)}^{-N}{\parallel }_q<\mathrm{\infty }$. If $p=\mathrm{\infty }$, choose $N>n+1$ and use any finite essential bound for $\mathrm{\mid }f\mathrm{\mid }$. The same estimates on bounded balls give local integrability of the chosen representatives.

For $n\ge 1$, as a boundary calculation, define $f_r(x)=\mathrm{\mid }x{\mathrm{\mid }}^r$ for $x\mathrm{\ne }0$ and assign any finite value, say $f_r(0)=0$, when $r<0$; set $f_0\equiv 1$, and for $r>0$ use the usual value $f_r(0)=0$. The value at this measure-zero point has no effect on local integrability or the induced distribution. The function $f_r$ is locally integrable at the origin exactly when $r>-n$: for $r<0$, the dyadic annuli $2^{-j-1}\le \mathrm{\mid }x\mathrm{\mid }<2^{-j}$ give a series comparable to ${\sum }_j{2}^{-j(n+r)}$; for $r\ge 0$ there is no singularity, while the reverse bound on a fixed cone gives divergence when $r\le -n$. At infinity it has polynomial growth. Therefore $f_r$ is among the tempered regular examples precisely for the locally meaningful range $r>-n$.