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 _ = ₈=₁ⁿ (² xᵢ² + 2 ᵢ xᵢ) where = (₁, , ₙ) Rⁿ\0\. The operator _ is selfadjoint with respect to the measure d_ (x) =e^2, xdx. This measure is not doubling but it is locally doubling in Rⁿ. We define, for every M>0 and k N, the operators Wᵏ, ₌, * (f) = ₓ>₀|Aᵏ, ₌, ₓ (f) |, 5mmg, ₌ᵏ (f) = (₀^|Aᵏ, ₌, ₓ (f) |²dtt) ^1{2}, \, k 1, the -variation operator V_ (\Aᵏ, ₌, ₓ\ₓ>₀) (f) = ₀₂, ₀₍₃, ₈₅ \ₓ䲛\₉ ₍ is a decreasing sequence in (0, ), the oscillation operator O (\{A, ₌, ₓᵏ\ₓ>₀, \tⱼ\₉ ₍) (f) = (₉ ₍\;\;ₓ_₉+₁ 0. We denote by T, ₌ᵏ any of the above operators. We analyze the boundedness of Tᵏ, ₌ on Lᵖ (Rⁿ, _) into itself, for every 1<p<, and from L¹ (Rⁿ, _) into L^{1, (Rⁿ, _). In addition, we obtain boundedness properties for the operator G, ₌^k, , 1 <2M, defined by G, ₌^k, (f) = (₀^|t^ /2+k ₜᵏD^ () (I-t _) ^-M (f) |²dtt) ^1{2}, for certain differentiation operator D^ ().