ARAZY'S CONJECTURE CONCERNING SCHUR MULTIPLIERS: REVISITED AND RESOLVED

Abstract

Let 𝒮^r denote the Schatten--von Neumann class and let S_{Ψ_f,λ} be the Schur--Hadamard multiplier whose symbol is the divided-difference matrix of f along λ. Let 0<α,r<∞, let f∈ C¹([−1,1]) satisfy f(0)=0, |f'(t)|≲ |t|^α, and let λ∈ℓ^r be real with |λ|_ℓ^∞≤1. We determine the pairs 0<p,q≤∞ for which S_{Ψ_f,λ}:𝒮^q→𝒮^p is bounded for every such f and λ. Our principal new positive estimates treat q=∞,1 and 0<p<1. Precisely, we show that the boundedness holds exactly when 1/p≤α/r +min {1,1/q}. In particular, this resolves the untreated cases in [Arazy, PAMS, 1982] and [Potapov, Sukochev, Tomskova, Adv. Math., 2015]. Our method also delivers a new proof of the main results in just cited papers.

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BibTeX RIS

BibTeX

@misc{huang2026arazys,
  title = {{ARAZY'S CONJECTURE CONCERNING SCHUR MULTIPLIERS: REVISITED AND RESOLVED}},
  author = {Jinghao Huang and Fedor Sukochev},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17144},
  eprint = {2609.17144},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05740214v1},
}

RIS

TY  - UNPB
TI  - ARAZY'S CONJECTURE CONCERNING SCHUR MULTIPLIERS: REVISITED AND RESOLVED
AU  - Jinghao Huang
AU  - Fedor Sukochev
PY  - 2026
DA  - 2026/09/07
PB  - HAL
DO  - 10.48550/arxiv.2609.17144
UR  - https://hal.science/hal-05740214v1
AB  - Let 𝒮^r denote the Schatten--von Neumann class and let S_Ψ_f,λ be the Schur--Hadamard multiplier whose symbol is the divided-difference matrix of f along λ. Let 0<α,r<∞, let f∈ C^1([−1,1]) satisfy f(0)=0, |f'(t)|≲ |t|^α, and let λ∈ℓ^r be real with |λ|_ℓ^∞≤1. We determine the pairs 0<p,q≤∞ for which S_Ψ_f,λ:𝒮^q→𝒮^p is bounded for every such f and λ. Our principal new positive estimates treat q=∞,1 and 0<p<1. Precisely, we show that the boundedness holds exactly when 1/p≤α/r +min {1,1/q}. In particular, this resolves the untreated cases in [Arazy, PAMS, 1982] and [Potapov, Sukochev, Tomskova, Adv. Math., 2015]. Our method also delivers a new proof of the main results in just cited papers.
ER  -

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