Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.13275

Block Repetition of Numerical Invariants for the Submodules $[z^k-w^k]$ in $H^2(\mathbb D^2)$

Yin Liu·Yufeng Lu·Chao Zu
View original source ↗Read source document ↗Export metadata ↓

Abstract

For $k\ge 2$, let $M_k=[z^k-w^k]$ be the principal homogeneous submodule of the Hardy space over the bidisk. We determine Yang's complete sequence of numerical invariants and prove $$ Σ_0(M_k)=\frac{π^2}{6},\qquad Σ_j(M_k)=Σ_{\lceil j/k\rceil}([z-w]),\quad j\ge1. $$ The proof exploits a residue-class decomposition of the Toeplitz matrices associated with the graded wandering spaces. Consequently, Yang's monotonicity conjecture holds for the family $\{M_k:k\ge2\}$. We also show that the nonzero spectral data of the core operator are independent of $k$, whereas the numerical invariant sequence recovers $k$ from the length of its constant blocks. Thus the higher numerical invariants detect module-theoretic information invisible to the core spectrum.

Explore related subjects

math.FAmath.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yin Liu, Yufeng Lu, Chao Zu. 2026-08-13. Block Repetition of Numerical Invariants for the Submodules $[z^k-w^k]$ in $H^2(\mathbb D^2)$. https://arxiv.org/abs/2608.13275

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Conditional expectation operators on $C(X)$

At the COSAEF conference in 2021, several participants asked the question whether a conditional expectation operator in the sense of Kuo, Labaushagne and Watson could be constructed in vector lattices other than $\mathcal{L}_p$ spaces and in particular in $C(X)$. This work answers positively to this question and participates in an old discussion on integrals in $C(X)$ space.

math.FA↗
arXiv2026

Fixed Point Rigidity of the Operator $Γ_pΠ_p^\ast$ and the LYZ Conjecture

We characterize the fixed points of the operator $Γ_pΠ_p^\ast$ for $n\geq 3$ and $1 0$ if and only if $K$ is an origin-centered ellipsoid, thereby settling the Lutwak--Yang--Zhang fixed-point conjecture in this range. Our proof is based on a variational analysis along linear reflection shadow systems. To address the nonlinear structure of the $L_p$ setting, we introduce the $L_p$-Projection Rolodex, which provides a dimensional reduction of the volume of the polar $L_p$-projection body to weighted lower-dimensional sectional functionals. A suitable change of variables, together with Ball's harmonic Prékopa--Leindler inequality, yields the convexity needed along the shadow system. Under the fixed-point condition, a first-variation identity then forces $\operatorname{vol}_n(Π_p^\ast K_t)$ to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.

math.FA↗
arXiv2026

Logarithmic oscillatory multipliers and log-subdyadic square functions

We develop square-function estimates for Fourier multipliers whose local oscillation scale is \[ ρ(R)=\frac{R}{(\log R)^{γ-1}}, \qquad γ>1. \] This scale lies strictly between the dyadic scale and every fixed power-subdyadic scale at high frequency. For high-frequency symbols satisfying a localized Sobolev condition on balls of radius comparable to $ρ(R)$, we prove a pointwise square-function estimate and a weighted $L^2$ multiplier inequality. After adjoining a smooth compactly supported low-frequency part, we derive unweighted $L^p$ bounds. The weighted estimate is governed by a logarithmic geometric maximal operator which is strongly bounded above the critical $L^r$ threshold, satisfies weak type at the critical equality, and fails even weak type below it. As a model application, consider \[ L(ξ)=\frac12\log(e^2+|ξ|^2), \qquad m_{γ,β}(ξ)=L(ξ)^{-β}e^{iL(ξ)^γ}. \] For $p=2$, the associated multiplier is bounded on $L^2$ for every $β\geq0$. For $1 d(γ-1)\left|\frac12-\frac1p\right|. \] At the critical equality we obtain the corresponding Lorentz endpoint estimates.

math.FA↗