The sharp vector Riesz inequality for multidimensional potential fields
For an integer $d \ge 2$ and a real-valued $f \in C_c^\infty(\mathbb{R}^d)$, let $R_1 f, \dots, R_d f$ be the Riesz transforms, $$\widehat{R_j f}(\xi) = -i \frac{\xi_j}{|\xi|} \widehat f(\xi), \qquad \widehat f(\xi) = \int_{\mathbb{R}^d} e^{-i x \cdot \xi} f(x) \, dx,$$ and put $p^* = \max\{p, p/(p-1)\}$. Iwaniec and Martin proved that each $R_j$ has norm exactly $\cot(\pi/(2p^*))$ on $L^p(\mathbb{R}^d)$, $1 < p < \infty$, so the constant in the inequality below cannot be smaller. Bañuelos and Wang proved the inequality below with the constant $2(p^* - 1)$. For $p = 2$ both sides are equal, by Plancherel's theorem. Bañuelos proved that, for each $d$, the best constant divided by $\cot(\pi/(2p^*))$ tends to $1$ as $p \to \infty$ (2026).
Problem. Is $$\Bigl\| \Bigl( \sum_{j=1}^{d} |R_j f|^2 \Bigr)^{1/2} \Bigr\|_{L^p(\mathbb{R}^d)} \le \cot\!\Bigl( \frac{\pi}{2p^*} \Bigr) \|f\|_{L^p(\mathbb{R}^d)}$$ for every integer $d \ge 2$, every $1 < p < \infty$ and every real-valued $f \in C_c^\infty(\mathbb{R}^d)$?
Source: Bañuelos (2010), Problem 6, who conjectures the constant cot(π/(2p*)); see Bañuelos and Osękowski (2015).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CA — Classical Analysis and ODEs
- Secondary
- FA, PR
- Keywords
- AIM Open Applied Problems, Riesz transforms, sharp constants, singular integrals
References
@article{banuelos2010,
author = {Ba{\~n}uelos, Rodrigo},
title = {The foundational inequalities of {D}. {L}. {B}urkholder and some of their ramifications},
journal = {Illinois J. Math.},
volume = {54},
number = {3},
year = {2010},
pages = {789--868},
doi = {10.1215/ijm/1336049979},
eprint = {1012.4850},
archivePrefix = {arXiv},
}
@article{banuelos2015,
author = {Ba{\~n}uelos, Rodrigo and Os{\k e}kowski, Adam},
title = {Sharp martingale inequalities and applications to {R}iesz transforms on manifolds, {L}ie groups and {G}auss space},
journal = {J. Funct. Anal.},
volume = {269},
number = {6},
year = {2015},
pages = {1652--1713},
doi = {10.1016/j.jfa.2015.06.015},
eprint = {1305.1492},
archivePrefix = {arXiv},
}
@article{iwaniec1996,
author = {Iwaniec, Tadeusz and Martin, Gaven},
title = {Riesz transforms and related singular integrals},
journal = {J. Reine Angew. Math.},
volume = {473},
year = {1996},
pages = {25--58},
doi = {10.1515/crll.1995.473.25},
}
@article{banuelos1995,
author = {Ba{\~n}uelos, Rodrigo and Wang, Gang},
title = {Sharp inequalities for martingales with applications to the {B}eurling-{A}hlfors and {R}iesz transforms},
journal = {Duke Math. J.},
volume = {80},
number = {3},
year = {1995},
pages = {575--600},
doi = {10.1215/S0012-7094-95-08020-X},
}
@misc{banuelos2026,
author = {Ba{\~n}uelos, Rodrigo},
title = {Cotlar martingale transforms and related singular integrals},
year = {2026},
eprint = {2604.09365},
archivePrefix = {arXiv},
}
@misc{aim-vector-riesz-sharp-norm,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The sharp vector Riesz inequality for multidimensional potential fields}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/426-vector-riesz-sharp-norm.md},
note = {Entry 426 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00399, "The sharp vector Riesz inequality for multidimensional potential fields", clopen.solutions.