[论文解读] Riesz transforms associated with the Grushin operator with drift

Nishta Garg, Rahul Garg|arXiv (Cornell University)|Mar 16, 2026

Advanced Harmonic Analysis Research被引用 0

一句话总结

本论文通过从海森堡-赖特群结果的转移及指数测度的对称性，建立了带漂移的 Grushin 算子的一阶及高阶 Riesz 变换的强类型 (p,p) 与弱类型 (1,1) 有界性。

ABSTRACT

We consider the Grushin operator with drift which is symmetric with respect to a measure having exponential growth. For the corresponding Riesz transforms, we study strong-type $(p, p)$, $1 < p < \infty$, and weak-type $(1, 1)$ boundedness.

研究动机与目标

• Motivate and study Riesz transforms of arbitrary order for the Grushin operator with drift that is symmetric w.r.t. an exponential growth measure.
• Extend known Lp boundedness results to the drifted Grushin setting and analyze endpoint weak-type properties.
• Leverage transference techniques to relate Grushin problems to drifted sub-Laplacians on Heisenberg–Reiter groups.

提出的方法

• Define the Grushin operator with drift G_a as G_a = G - 2 a · ∇_{x'}, symmetric on L^2(R^{n+m}, dμ_a).
• Express Riesz transforms R_{α,a} = X^α G_a^{−|α|/2} via functional calculus and heat semigroups.
• Relate G_a to the sub-Laplacian with drift on Heisenberg–Reiter groups and apply Coifman–Weiss transference to transfer Lp-boundedness.
• Use unitary representation to connect Grushin gradient fields with Heisenberg–Reiter gradients.
• Employ asymptotics of ball volumes and heat kernel bounds to control kernels and apply endpoint arguments.
• Show convergence (after scaling and translation) to Laplacian with drift to deduce endpoint behavior.

实验结果

研究问题

• RQ1 What Lp boundedness do the first and higher order Riesz transforms R_{α,a} have with respect to dμ_a for 1<p<∞?
• RQ2 Are the first order Riesz transforms of G_a weak-type (1,1) on R^{n+1} with uniform bounds in a?
• RQ3 Do all Riesz transforms of order k≥3 fail to be of weak-type (1,1) with respect to dμ_a?
• RQ4 How does the Grushin problem with drift relate to drifted sub-Laplacians on Heisenberg–Reiter groups via transference?
• RQ5 Can endpoint results for the drifted Grushin operator be deduced from the corresponding Laplacian with drift on R^{n+m}?

主要发现

• For any multi-index α with |α|=k≥1 and 1<p<∞, the Riesz transform R_{α,a} is bounded on L^p(R^{n+m}, dμ_a), uniformly in a.
• On R^{n+1} (i.e., m=1), the first-order Riesz transforms are of weak-type (1,1) with respect to dμ_a, uniformly in a.
• For k≥3, not all Riesz transforms of order k are of weak-type (1,1) with respect to dμ_a.
• The boundedness results for Riesz transforms of G_a follow from known results for the sub-Laplacian with drift on Heisenberg–Reiter groups via Coifman–Weiss transference.
• Proposition establishing that a Riesz transform for Grushin with (scaled) drift, under conjugation by translations and dilations, converges to the Riesz transform for the Laplacian with drift on R^{n+m}.
• From this convergence, endpoint weak-type results for G_a follow by comparison to the Laplacian with drift results in Li–Sjögren–Wu and Li–Sjögren.