Percentile oscillations and the intrinsic square function
Abstract
Wilson's intrinsic square function tests the input against all the cancellative functions of a given smoothness, at every location and scale. We show that it is dominated pointwise by a sparse square function, whose coefficients are the percentile oscillations of the input modulo polynomials, at every polynomial order below the smoothness plus one. These coefficients are local sizes, which nevertheless retain essentially all the cancellation: the weighted norms of the sparse square function are equivalent to smooth grand maximal norms, for all 0<p<∞ and A∞ weights, with explicit dependence on the weight. These are weighted Hardy norms in the range allowed by the smoothness. Consequences include potential bounds, and weighted Hardy space estimates with the sharp factor [w]A∞1/2. For the derivatives of Calderón--Zygmund operators, we characterize the corresponding square function estimates by testing conditions on the polynomial symbols of their wavelet resolution. In the quadratic setting, the factor [w]A∞1/2 detects a transition from BMO testing conditions to vanishing symbols.
Related
- BMO-Type characterizations and endpoint estimates for commutators of intrinsic square functions on Orlicz-Hardy spaces
- On the counting function of square-full numbers
- On the weighted weak-type constant for the dyadic square function
- The square function estimates for cones in arbitrary dimensions and elliptic quadrics
- The square function estimate for the cone in four dimensions
BibTeX
@misc{plinio2026percentile,
title = {{Percentile oscillations and the intrinsic square function}},
author = {Francesco Di Plinio},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01687},
eprint = {2611.01687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01687},
}
RIS
TY - UNPB
TI - Percentile oscillations and the intrinsic square function
AU - Francesco Di Plinio
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01687
UR - https://arxiv.org/abs/2611.01687
AB - Wilson's intrinsic square function tests the input against all the cancellative functions of a given smoothness, at every location and scale. We show that it is dominated pointwise by a sparse square function, whose coefficients are the percentile oscillations of the input modulo polynomials, at every polynomial order below the smoothness plus one. These coefficients are local sizes, which nevertheless retain essentially all the cancellation: the weighted norms of the sparse square function are equivalent to smooth grand maximal norms, for all 0<p<∞ and A_∞ weights, with explicit dependence on the weight. These are weighted Hardy norms in the range allowed by the smoothness. Consequences include potential bounds, and weighted Hardy space estimates with the sharp factor [w]_A_∞^1/2. For the derivatives of Calderón--Zygmund operators, we characterize the corresponding square function estimates by testing conditions on the polynomial symbols of their wavelet resolution. In the quadratic setting, the factor [w]_A_∞^1/2 detects a transition from BMO testing conditions to vanishing symbols.
ER -