Proposition 3

Let N be a norm on $\mathbf{R}^n,r$ a real number and $p\in [1,+\infty [$.

a) The function $(1 + N)^r$ belong to $\mathscr{L}^p(\mathbf{R}^n, \mu)$ if and only if $rp <-n$;

a$')$ The restriction to $\mathbf{Z}^n$ of the function $(1 + N)^r$ belong to $\ell^p(\mathbf{Z}^n)$ if and only if $rp <-n$;

b) Let V be a bounded measurable neighbourhood of 0 in $\mathbf{R}^n$. The function $N^r$ belong to $\mathscr{L}^p(\mathbf{R}^n-V, \mu)$ if and only if $rp <-n$.

By virtue of the equivalence of norms on $\mathbf{R}^n$ (EVT I, p. 14, th. 2 and TG, IX, p. 32, prop. 8), we may suppose that the norm N is given by $N(x) =$ sup$|x_i|$ for $x= (x_i)\in \mathbf{R}^n$. Let B denote the unit ball of $\mathbf{R}^n$ for this norm.

Let V be a bounded measurable neighbourhood of 0 in $\mathbf{R}^n$. The function $N^r$ is continuous and bounded on (B- $V$)$\cup (V$- B), which shows that assertion b) is valid if and only if it is so when V = B, which we shall henceforth suppose. Since the function $(1 + N)^r$ is continuous and bounded on B, and satisfies the inequalities $N^r\leqslant (1 + N)^r\leqslant 2^rN^r$ on $\mathbf{R}^n-$ B, we see that assertions a) and b) are equivalent.

Let us prove b) when V = B. We may suppose that $p= 1$. Since the case where $r >0$ is elementary, suppose that $r\leqslant 0$. For every integer $j\geqslant 1$, the set

is of measure $2^{nj}(2^n-1)$. The sets $(C_j)_{j\geqslant 1}$ form a partition of $\mathbf{R}^n-$ B. According to INT, V, p. 4, § 1, No.$^o1$, cor., one therefore has

$\mathbf{R}^{n-}Bj\geqslant 1C_j$

$\leqslant \sum_{j\geqslant 1}2^{n+nj}2^{r(j-1)}= 2^{n-r}\sum_{j\geqslant 1}2^{(n+r)j}$ which is finite if $r <-n$. On the other hand, one has (loc. cit.)

$\mathbf{R}^{n-}Bj\geqslant 1j\geqslant 1$

which is infinite if $r\geqslant -n$.

One proves a$')$ similarly by considering the sets $C_j\cap \mathbf{Z}^n$ which cover $\mathbf{Z}^n-\{0\}$ and satisfy

Card(C$_j\cap \mathbf{Z}^n$) $= (2^{j+1}-1)^n-(2^j-1)^n$,

which belongs to the interval $[(1-2^{-n})(2^{j+1}-1)^n,2^{n(j+1)}]$.