p>2时广义希尔伯特算子在H^p上的向量卡尔松、卡尔德隆与泊松原子判据
Vector-Carleson, Calderón, and Poisson-Atomic Criteria for Generalized Hilbert Operators on $H^p$, $p>2$
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中文总结 AI 辅助
该研究针对p>2时广义希尔伯特算子在H^p上的有界性，建立了向量卡尔松等判据，构造了特殊符号并证明泊松原子检验定理，给出原子复杂度公式与检验界。
中文摘要 AI 辅助
设ℋ_g f(z)=∫₀¹ f(t)g’(tz)dt，且2<p<∞。令t=2p/(p-2)，X_j=2^{-j/p'}Δ_j g'，其中Δ_j为硬二进泰勒投影。我们证明：ℋ_g:H^p→H^p有界当且仅当映射h↦(X_j h)_{j≥0}从H^t到ℓ^t(H^2)有界。该嵌入范数的平方等于正列算子b↦∑_j b_j|X_j|²从ℓ^{p/2}到L^{p/2}的范数。坐标尾部给出本质范数估计与精确紧性判据，而哈代对偶性给出等价的仿积公式。该判据在容许解析二进分解下具有定量不变性，且定义了与分解无关的卡尔德隆符号空间，其等于希尔伯特范围乘子空间。我们构造了一个有界非紧密频率符号，它不属于已知的分块充分类。我们还证明了一个精确泊松原子检验定理：有限正泊松混合物可恢复全部范数，但固定原子数不足以做到。最后，聚合概率密度给出内在原子复杂度公式与有限带宽检验界。
英文摘要
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 $X_j=2^{-j/p'}Δ_j g'$, where $Δ_j$ 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 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.
