Proposition 21

Let N be the euclidean norm on $\mathbf{R}^n$. Let $k$ be an integer $\geqslant 0$. The Sobolev space $H^k(\mathbf{R}^n)$ is the space of $f\in \mathscr{S}'(\mathbf{R}^n)$ such that $(1 + N^k)\mathscr{F}(f)$ belongs to $L^2(\mathbf{R}^n)$.

We proceed by induction on $k$. When $k= 0$, the result is a consequence of the Plancherel theorem (II, p. 215, th. 1). Suppose that $k= 1$. For $f\in \mathscr{S}'(\mathbf{R}^n)$, one has $(1 + N)\mathscr{F}f\in L^2(\mathbf{R}^n)$ if and only if $f\in L^2(\mathbf{R}^n)$ and $N\mathscr{F}f\in L^2(\mathbf{R}^n)$. Moreover, $N\mathscr{F}\in L^2(\mathbf{R}^n)$ if and only if, for every $\alpha \in \mathbf{N}^n$ such that $|\alpha |= 1$, one has $X^{\alpha}\mathscr{F}f\in L^2(\mathbf{R}^n)$. Since one has $\mathscr{F}(\partial^{\alpha}f) = 2i\pi X^{\alpha}\mathscr{F}f$, this condition means that $\mathscr{F}(\partial^{\alpha}f)\in L^2(\mathbf{R}^n)$ for every $\alpha$ with $|\alpha |= 1$, that is, $\partial^{\alpha}f\in L^2(\mathbf{R}^n)$ for every $\alpha$ with $|\alpha |= 1$. It follows that the assertion is true for $k= 1$.

Suppose now that $k\geqslant$ 2 and that the assertion concerning $H^{\ell}(\mathbf{R}^n)$ is valid for every positive integer $\ell \leqslant k-1$. Let $f\in \mathscr{S}'(\mathbf{R}^n)$. By definition, one has $f\in H^k(\mathbf{R}^n)$ if and only if $f\in L^2(\mathbf{R}^n)$ and, for every $\beta \in \mathbf{N}^n$ such that $|\beta |\leqslant 1$, the distribution $\partial^{\beta}f$ belongs to $H^{k-1}(\mathbf{R}^n)$. This is equivalent, by the induction hypothesis, to $f\in L^2(\mathbf{R}^n)$ and $(1 + N^{k-1})\mathscr{F}(\partial^{\beta}f)\in L^2(\mathbf{R}^n)$ for every $\beta \in \mathbf{N}^n$ such that $|\beta |\leqslant 1$. Since $\mathscr{F}(\partial^{\beta}f) = (2i\pi X)^{|\beta|}\mathscr{F}(f)$, the condition $f\in H^k(\mathbf{R}^n)$ is equivalent to saying that $\mathscr{F}f\in L^2(\mathbf{R}^n)$ and $(1 + N^{k-1})X^{\beta}\mathscr{F}f\in L^2(\mathbf{R}^n)$ for $\beta \in \mathbf{N}^n$ such that $|\beta |\leqslant 1$.

The inequalities

then imply that $f\in H^k(\mathbf{R}^n)$ if and only if $(1+N^k)\mathscr{F}\in L^2(\mathbf{R}^n)$.