Riesz transforms associated with the twisted Laplacian with drift

We consider the Riesz transforms of arbitrary order associated with the twisted Laplacian with drift on $\mathbb{C}^n$ and study their strong-type $(p, p)$, $1<p<\infty$, and weak-type $(1, 1)$ boundedness.

💡 Research Summary

The paper investigates Riesz transforms of arbitrary order associated with the twisted Laplacian on ℂⁿ when a non‑zero drift vector ν∈ℝⁿ×ℝⁿ is present. The twisted Laplacian L_λ (λ∈ℝ{0}) is defined by conjugating the sub‑Laplacian on the Heisenberg group Hⁿ with the phase factor e^{iλs}; explicitly L_λ=Δ_z−(λ²/4)|z|²−iλN, where Δ_z is the Euclidean Laplacian on ℂⁿ and N is the rotation operator. Adding the drift yields the operator

 L_{ν,λ}=L_λ+2ν·∇(λ), ∇(λ)=(X₁(λ),…,X_n(λ),Y₁(λ),…,Y_n(λ)),

with X_j(λ)=∂{x_j}+i(λ/2) y_j and Y_j(λ)=∂{y_j}−i(λ/2) x_j. The measure

 dμ_ν(x,y)=e^{2(a·x+b·y)}dxdy

(with ν=(a,b)) makes L_{ν,λ} essentially self‑adjoint and negative definite on L²(ℂⁿ,dμ_ν).

The authors consider monomials P_k(λ) of total degree k in the vector fields X_j(λ),Y_j(λ) and define the k‑th order Riesz transform

 R_{k,ν,λ}=P_k(λ)(−L_{ν,λ})^{−k/2}.

The main contributions are four theorems.

Theorem 1.5 (Strong Lᵖ boundedness). For every 1<p<∞ and any integer k≥1, R_{k,ν,λ} extends to a bounded operator on Lᵖ(ℂⁿ,dμ_ν) with a norm that does not depend on λ or ν. The proof relies on the explicit heat kernel

 p_{t,λ}(z,w)=e^{−|ν|²t}e^{−a·(x+u)−b·(y+v)}p_{t,λ}(z−w)e^{iλ/2 Im(z·\bar w)}

and standard Calderón–Zygmund theory after verifying size and smoothness estimates for the kernel of (−L_{ν,λ})^{−k/2}.

Theorem 1.6 (Endpoint weak‑type estimates). By rotating the drift to ν=e₁ (the first coordinate vector) the authors isolate the total power k₁ contributed by X₁(λ) and Y₁(λ) in the monomial.

• If k₁∈{0,1,2}, the transform R_{k,e₁,λ} is of weak‑type (1,1) uniformly in λ.
• If k₁≥3, a logarithmic correction is necessary:

 μ_{e₁}{|R_{k,e₁,λ}f|>α} ≤ C_k ∫_{ℂⁿ} (|f|/α)