The product formula of theorem 15.30.6 identifies a complete subspace of density roots with uniformly vanishing tails even when the ambient lower mass is zero. Use the product data $X_0,Y,N,H,c_0,\mu ,R$ of 15.30.5. The spaces $\mathcal D_ H$ and $\mathcal D_ H^0$ are now taken on the $N$-dimensional product with reference volume $\mathrm dV$.
Proposition 15.32.11.7A2CC1A3258E For real or complex $\xi \in L^ N(Y,\kappa ^ k)$, put $J\xi =R^{1/N}\pi _ Y^*\xi $. Then
\[ \| J\xi \| _{L^ N(\mathrm dV)}=\| \xi \| _{L^ N(\kappa ^ k)},\qquad \rho _ H(J\xi )=c_0^{-1/N}\| \xi \| _{L^ N(\kappa ^ k)}. \]
The image of $J$ is a closed linear subspace of $\mathcal D_ H^0$. If $g=|\xi |^ N$ and $A=\max _ XR$, then
\[ \rho _ H\bigl (J\xi \mathbf1_{\{ |J\xi |>K\} }\bigr ) \leq \left(\frac1{c_0}\int _{\{ g>K^ N/A\} }g\kappa ^ k\right)^{1/N} \longrightarrow 0. \]
For $\mathrm dV=\mu $, so that $R=1$, the exact tail formula is
\[ \rho _ H\bigl (\pi _ Y^*\xi \mathbf1_{\{ |\xi |>K\} }\bigr ) =\left(\frac1{c_0}\int _{\{ |\xi |>K\} }|\xi |^ N\kappa ^ k\right)^{1/N}. \]
Proof.
Since $|J\xi |^ N=R|\xi |^ N$, integration against $\mathrm dV$ gives the first norm identity. For nonzero $\xi $, theorem 15.30.6 applied to $g=|\xi |^ N$ gives the root-functional identity; for $\xi =0$ it holds by definition. Applying the two identities to differences yields
\[ \| J\xi -J\eta \| _{\mathcal D_ H} =(1+c_0^{-1/N})\| \xi -\eta \| _{L^ N(\kappa ^ k)}. \]
Bounded approximations of $\xi $ in $L^ N$ have bounded images because $R$ is bounded, so $J\xi \in \mathcal D_ H^0$. Completeness of $L^ N(Y,\kappa ^ k)$ and the displayed identity make the image closed.
On $\{ |J\xi |>K\} $ one has $g>K^ N/A$, and hence
\[ |J\xi |\mathbf1_{\{ |J\xi |>K\} } \leq \bigl (Rg\mathbf1_{\{ g>K^ N/A\} }\bigr )^{1/N}. \]
Monotonicity of the root functional and its identity for factor roots give the estimate, with the value zero when the tail vanishes. Integrability of $g$ makes the bound tend to zero; the $L^ N$ tails also vanish. When $R=1$, the amplitude cutoff itself is pulled back from $Y$, so the same norm identity gives equality.
$\square$
For $Y=\mathbb P^ k$ and $\mathrm dV=\mu $, take $p_\alpha =\pi _ Y^*(q_\alpha ^{1/N})$ with $q_\alpha $ from lemma 15.30.1. Then $p_\alpha \in \mathcal D_ H^0$ but $p_\alpha \notin L^ r(\mathrm dV)$ for every $r>N$. Moreover,
\[ \rho _ H\bigl (p_\alpha \mathbf1_{\{ p_\alpha >K\} }\bigr ) \asymp (\log K)^{-k(1-\alpha )/N}\qquad (K\to +\infty ). \]
Indeed, the radial estimate of that lemma gives $q_\alpha \asymp r^{-k}(1-\log r)^{-1-k(1-\alpha )}$. With $L=1-\log r$, the density level set $\{ q_\alpha >K^ N\} $ lies between two sets $\{ L>c\log K\} $ for fixed positive constants $c$, once $K$ is large. Integrating the radial density $L^{-1-k(1-\alpha )}\, \mathrm dL$ and taking its $N$th root gives the claimed tail. The integrability exclusion follows from $q_\alpha \notin L^ s$ for $s>1$.
These roots enlarge the examples in $\mathcal D_ H^0$ beyond the subcritical $L^ r$ classes. They give neither $\mathcal D_ H^0=\mathcal D_ H$ nor a general criterion for an exact equation at zero lower mass.
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