On the multidimensional Nazarov lemma

TL;DR

This work proves a multidimensional analogue of Nazarov's lemma, establishing that for a positive function $\Omega$ in $L^1(dP_n)\cap \mathrm{Lip}(\mathbb{R}^n)$ and any $\varepsilon>0$ there exists a majorant $\Omega_1$ with $\Omega\le\Omega_1$, $\Omega_1\in L^1(dP_n)$, and all Riesz transforms $R_j\Omega_1$ belonging to $\mathrm{Lip}(\varepsilon,\mathbb{R}^n)$. The proof develops a generalized regularized system of dyadic squares and tails, mirroring the one-dimensional construction but with intricate multi-parameter geometry and Hadamard–Landau-type estimates to control the majorant and the Riesz transforms. The local-to-global strategy uses a local Nazarov lemma on squares and a dyadic tiling to assemble a global majorant with dimension-dependent constants, and the Appendix extends the result to arbitrary dimension with the corresponding $R_j$ operators. This provides a crucial tool for harmonic analysis in multiple dimensions, with implications related to Beurling–Mallliavin-type results and spectral majorization, while noting the method does not straightforwardly extend to the double Hilbert transform.

Abstract

In this article we prove a multidimensional version of the Nazarov lemma. The proof is based on an appropriate generalisation of the regularised system of intervals introduced by Havin, Nazarov and Mashreghi to several dimensions.

Theorems & Definitions (15)

• Theorem A
• Definition 1
• Definition 2
• Remark 1
• Remark 2
• Theorem 1
• Remark 3
• Lemma 1
• Remark 4
• Definition 3