Lᵖ-boundedness properties for some harmonic analysis operators defined by resolvents for a Laplacian with drift in Euclidean spaces

We consider the Laplacian with drift in Rⁿ defined by Δ_ν = ∑ᵢ₌₁ⁿ(∂²/∂ xᵢ² + 2 νᵢ∂ ∂xᵢ) where ν=(ν₁,…,νₙ)∈ Rⁿ\0\. The operator Δ_ν is selfadjoint with respect to the measure dμ_ν(x)=e2ν,xdx. This measure is not doubling but it is locally doubling in Rⁿ. We define, for every $M>0$ and k ∈ N, the operators Wᵏν,M,*(f) = t>0|Aᵏν,M,t(f)|,5mmgν,Mᵏ(f) = (∫₀^∞|Aᵏν,M,t(f)|²dt/t)1/2,\,k≥ 1, the ρ-variation operator $$ V_ρ( \{A^kν,M,t\}t>0)(f)= 0<t_1<⋯<t_,\, ∈ N(∑⁻¹ⱼ₌₁|A^kν,M,t_j(f)- A^k_{ν,M,tⱼ₊₁}(f)|^ρ)1/ρ,\;\; ρ>2, $$ and, if $\{t_j\}j∈ N$ is a decreasing sequence in $(0,∞)$, the oscillation operator $$ O(\{Aν,M,t^k\}t>0,\{t_j\}j∈ N)(f)=(∑j∈ N\;\;_{tⱼ₊₁≤ ε <ε '≤ t_j}|A^kν,M,ε(f)-A^kν,M,ε '(f)|^2 )1/2. $$ where $A^kν,M,t=t^k∂^k_t(I-tΔ_ν)-M$, $t>0$. We denote by $Tν,M^k$ any of the above operators. We analyze the boundedness of $T^kν,M$ on $L^p( R^n,μ_ν)$ into itself, for every $1<p<∞$, and from $L^1( R^n,μ_ν)$ into $L1,∞( R^n,μ_ν)$. In addition, we obtain boundedness properties for the operator $Gν,Mk,$, $1≤ <2M$, defined by $$ Gν,Mk,(f)=(∫_0^∞|t/2+k∂ _t^kD⁽⁾(I-tΔ _ν)-M(f) |^2dt/t)1/2, $$ for certain differentiation operator $D⁽⁾$.