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

Abstract

Let ${\mathcal{H}}_{g}f(z)={\int }_0^{1}f(t)g^{\prime }(tz),dt$, and let $2<p<\infty$. Set $t=2p/(p-2)$ and $X_j=2^{-j/p^{\prime }}{\Delta }_j{g}^{\prime }$, 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 $h\mapsto (X_{j}h{)}_{j\ge 0}$ 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 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.

21 pages, no figures