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

Abstract

Let $\mathcal H_g f(z)=\int_0^1 f(t)g'(tz),dt$, and let $2<p<\infty$. Set $t=2p/(p-2)$ and $X_j=2^{-j/p'}\Delta_j g'$, where $\Delta_j$ is the hard dyadic Taylor projection. We prove that $\mathcal H_g:H^p\to H^p$ is bounded if and only if $h\mapsto(X_jh)_{j\geq0}$ is bounded from $H^t$ to $\ell^t(H^2)$. The square of this embedding norm equals the norm of the positive column operator $b\mapsto\sum_j b_j|X_j|^2$ from $\ell^{p/2}$ to $L^{p/2}$. 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 Calder\'on 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.