Vector-Carleson, Calderón, and Poisson-Atomic Criteria for Generalized Hilbert Operators on $H^p$, $p>2$

Let ${\mathcal{H}}_{g}f(z)={\int }_0^{1}f(t)g^{\mathrm{\prime }}(tz),dt$, and let $2<p<\mathrm{\infty }$. Set $t=2p\mathrm{/}(p-2)$ and $X_j=2^{-j\mathrm{/}{p}^{\mathrm{\prime }}}{\mathrm{\Delta }}_j{g}^{\mathrm{\prime }}$, where ${\mathrm{\Delta }}_j$ is the hard dyadic Taylor projection. We prove that ${\mathcal{H}}_g:H^p\to H^p$ is bounded if and only if $h\mapsto (X_{j}h{)}_{j\ge 0}$ is bounded from $H^t$ to ${\mathrm{\ell }}^t(H^2)$. The square of this embedding norm equals the norm of the positive column operator $b\mapsto {\sum }_j{b}_j\mathrm{\mid }{X}_j{\mathrm{\mid }}^2$ from ${\mathrm{\ell }}^{p\mathrm{/}2}$ to $L^{p\mathrm{/}2}$. Coordinate tails yield essential-norm estimates and an exact compactness criterion, while Hardy duality gives an equivalent paraproduct formulation. The criterion is quantitatively invariant under admissible analytic dyadic resolutions and defines a resolution-independent Calderón symbol space equal to the Hilbert-range multiplier space. We construct a bounded noncompact dense-frequency symbol outside the known blockwise sufficient class. We also prove an exact Poisson-atomic testing theorem: finite positive Poisson mixtures recover the full norm, but no fixed atom count suffices. Finally, aggregate probability densities give an intrinsic atomic-complexity formula and a finite-bandwidth testing bound.