Proposition 15.32.92210D4123A64

Proof. For $K>0$, set $p_ K=p\min (1,K/|p|)$ where $p\ne 0$, and $p_ K=0$ where $p=0$. Then $|p-p_ K|=(|p|-K)_+$ and

Since the $L^ n$ norms of the tails tend to zero, these inequalities and the monotonicity of $\rho _ h$ show that the second condition is equivalent to $p_ K\to p$ in $\mathcal D_ h$; in particular, it implies the first. Conversely, if $a$ is bounded and $K\geq \| a\| _{L^\infty }$, then $(|p|-K)_+\leq |p-a|$, so $\| p-p_ K\| _{\mathcal D_ h}\leq \| p-a\| _{\mathcal D_ h}$. Approximating $p$ by bounded functions $a$ shows that the first condition implies $p_ K\to p$, hence the second.

Assume the second condition, and let $q_ j$ be as in the third. Fix $K$. On $\{ |p|\leq K\} $, the functions $q_ j$ are bounded by $K$ and tend to zero in measure, hence in $L^{2n}$; by (15.32.8) with $r=2n$, the root functional of $q_ j\mathbf1_{\{ |p|\leq K\} }$ tends to zero. On the complementary set, $\rho _ h(q_ j\mathbf1_{\{ |p|>K\} })\leq \rho _ h(|p|\mathbf1_{\{ |p|>K\} })$. Letting first $j$ and then $K$ tend to infinity gives $\rho _ h(q_ j)\to 0$, and dominated convergence gives $\| q_ j\| _{L^ n}\to 0$. Conversely, the third condition applied to $q_ j=|p|\mathbf1_{\{ |p|>j\} }$ gives the second.

Since the second condition passes from $p$ to every function dominated by $|p|$, the space $\mathcal D_ h^0$ is an order ideal; it is closed by definition. For $p\in L^ r$ with $r>n$, the truncations $p_ K$ converge to $p$ in $L^ r$, hence in $\mathcal D_ h$ by (15.32.8). Bounded functions can be approximated in $L^{2n}$ by smooth functions, which then converge in $\mathcal D_ h$ by the same estimate; hence smooth functions are dense in $\mathcal D_ h^0$. Under positive lower mass, proposition 15.32.5 shows that $\mathcal D_ h=L^ n$ with an equivalent norm, and bounded functions are dense in $L^ n$; hence $\mathcal D_ h^0=\mathcal D_ h=L^ n$. $\square$