Vector-Carleson, Calderón, and Poisson-Atomic Criteria for Generalized Hilbert Operators on Hp, p>2
Abstract
Let ℋg f(z)=∫01 f(t)g'(tz),dt, and let 2<p<∞. Set t=2p/(p−2) and Xj=2−j/p'Δj g', where Δj is the hard dyadic Taylor projection. We prove that ℋg:Hp→ Hp is bounded if and only if h↦(Xjh)j≥0 is bounded from Ht to ℓt(H2). The square of this embedding norm equals the norm of the positive column operator b↦∑j bj|Xj|2 from ℓp/2 to Lp/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.
Related
- Complete mapping criteria for generalized Cesàro operators between Hardy spaces
- Poisson representations of Dunkl harmonic functions satisfying a Carleson condition
- The conformal Poisson embedding method for the anisotropic Calderón problem with a potential
- Carleson measures, Volterra integral operators and multipliers for QK spaces
- Wavelet resolution and Sobolev regularity of Calderón–Zygmund operators on domains
BibTeX
@misc{ma2026vectorcarleson,
title = {{Vector-Carleson, Calderón, and Poisson-Atomic Criteria for Generalized Hilbert Operators on $H^p$, $p>2$}},
author = {Yicen Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01116},
eprint = {2609.01116},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01116},
}
RIS
TY - UNPB
TI - Vector-Carleson, Calderón, and Poisson-Atomic Criteria for Generalized Hilbert Operators on H^p, p>2
AU - Yicen Ma
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01116
UR - https://arxiv.org/abs/2609.01116
AB - Let ℋ_g f(z)=∫_0^1 f(t)g'(tz),dt, and let 2<p<∞. Set t=2p/(p−2) and X_j=2^−j/p'Δ_j g', where Δ_j is the hard dyadic Taylor projection. We prove that ℋ_g:H^p→ H^p is bounded if and only if h↦(X_jh)_j≥0 is bounded from H^t to ℓ^t(H^2). The square of this embedding norm equals the norm of the positive column operator b↦∑_j b_j|X_j|^2 from ℓ^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.
ER -