Logarithmic oscillatory multipliers and log-subdyadic square functions

Abstract

We develop square-function estimates for Fourier multipliers whose local oscillation scale is ρ(R)=R/((log R)^γ−1), 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 L² 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)=1/2log(e²+|ξ|²), m_γ,β(ξ)=L(ξ)^−βe^{iL(ξ)^γ}. For p=2, the associated multiplier is bounded on L² for every β≥0. For 1<p<∞, p≠2, it is bounded on L^p under the sufficient condition β> d(γ−1)|1/2−1/p|. At the critical equality we obtain the corresponding Lorentz endpoint estimates.

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BibTeX RIS

BibTeX

@misc{vergara2026logarithmic,
  title = {{Logarithmic oscillatory multipliers and log-subdyadic square functions}},
  author = {Vicente Vergara},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.27746},
  eprint = {2605.27746},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.27746},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - Logarithmic oscillatory multipliers and log-subdyadic square functions
AU  - Vicente Vergara
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2605.27746
UR  - https://arxiv.org/abs/2605.27746
AB  - We develop square-function estimates for Fourier multipliers whose local oscillation scale is ρ(R)=R/((log R)^γ−1), 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)=1/2log(e^2+|ξ|^2), m_γ,β(ξ)=L(ξ)^−βe^iL(ξ)^γ. For p=2, the associated multiplier is bounded on L^2 for every β≥0. For 1<p<∞, p≠2, it is bounded on L^p under the sufficient condition β> d(γ−1)|1/2−1/p|. At the critical equality we obtain the corresponding Lorentz endpoint estimates.
ER  -

Abstract

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