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Product Pitt inequalities and signed Boas equivalences for general monotone functions
Abstract
We prove product-weight Pitt inequalities for real or complex functions on ℝd whose full mixed Vitali variation satisfies an exterior-tail general monotonicity condition. A two-parameter condition gives spatial weights with different powers near the coordinate hyperplanes and at infinity. For the classical parameters, a reverse estimate yields a Boas norm equivalence without sign, parity or separability assumptions. The equivalence holds for finite spatial norm and, for L1 data, characterizes finiteness of either norm; the unrestricted nonintegrable converse remains open. We also prove an equivalence under primitive control, give explicit examples, and establish sharpness of the Fourier-side exponent range.
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BibTeX
@misc{tokmagambetov2026product,
title = {{Product Pitt inequalities and signed Boas equivalences for general monotone functions}},
author = {Niyaz Tokmagambetov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41309},
eprint = {2610.41309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41309},
}
RIS
TY - UNPB
TI - Product Pitt inequalities and signed Boas equivalences for general monotone functions
AU - Niyaz Tokmagambetov
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41309
UR - https://arxiv.org/abs/2610.41309
AB - We prove product-weight Pitt inequalities for real or complex functions on ℝ^d whose full mixed Vitali variation satisfies an exterior-tail general monotonicity condition. A two-parameter condition gives spatial weights with different powers near the coordinate hyperplanes and at infinity. For the classical parameters, a reverse estimate yields a Boas norm equivalence without sign, parity or separability assumptions. The equivalence holds for finite spatial norm and, for L^1 data, characterizes finiteness of either norm; the unrestricted nonintegrable converse remains open. We also prove an equivalence under primitive control, give explicit examples, and establish sharpness of the Fourier-side exponent range.
ER -