$L^p$-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 ${\mathbb{R}}^n$ defined by ${\mathrm{\Delta }}_{\nu }={\sum }_{i=1}^n(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_i^2}+2{\nu }_i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i})$ where $\nu =({\nu }_1,\dots ,{\nu }_n)\in {\mathbb{R}}^n\setminus \{0\}$. The operator ${\mathrm{\Delta }}_{\nu }$ is selfadjoint with respect to the measure $d{\mu }_{\nu }(x)=e^{2⟨\nu ,x⟩}dx$. This measure is not doubling but it is locally doubling in ${\mathbb{R}}^n$. We define, for every $M>0$ and $k\in \mathbb{N}$, the operators $$W_{\nu ,M,\ast }^k(f)=\underset{t>0}{\sup }\mid A_{\nu ,M,t}^k(f)\mid ,g_{\nu ,M}^k(f)={\left({\int }_0^{\mathrm{\infty }}{\mid A_{\nu ,M,t}^k(f)\mid }^2\frac{dt}{t}\right)}^{\frac{1}{2}},k\ge 1,$$ the $\rho$-variation operator $$V_{\rho }(\{A_{\nu ,M,t}^k{\}}_{t>0})(f)=\underset{0<t_1<\cdots <t_{\mathrm{\ell }},\mathrm{\ell }\in \mathbb{N}}{\sup }{\left(\sum _{j=1}^{\mathrm{\ell }-1}{\mid A_{\nu ,M,t_j}^k(f)-A_{\nu ,M,t_{j+1}}^k(f)\mid }^{\rho }\right)}^{\frac{1}{\rho }},\rho >2,$$ and, if $\{t_j{\}}_{j\in \mathbb{N}}$ is a decreasing sequence in $(0,\mathrm{\infty })$, the oscillation operator $$O(\{A_{\nu ,M,t}^k{\}}_{t>0},\{t_j{\}}_{j\in \mathbb{N}})(f)=(\sum _{j\in \mathbb{N}}\underset{t_{j+1}\le \epsilon <{\epsilon }^{\mathrm{\prime }}\le t_j}{\sup }\mathrm{\mid }{A}_{\nu ,M,\epsilon }^k(f)-A_{\nu ,M,{\epsilon }^{\mathrm{\prime }}}^k(f){\mathrm{\mid }}^2{)}^{1\mathrm{/}2}\mathrm{.}$$ where $A_{\nu ,M,t}^k=t^k{\mathrm{\partial }}_t^k(I-t{\mathrm{\Delta }}_{\nu }{)}^{-M}$, $t>0$. We denote by $T_{\nu ,M}^k$ any of the above operators. We analyze the boundedness of $T_{\nu ,M}^k$ on $L^p({\mathbb{R}}^n,{\mu }_{\nu })$ into itself, for every $1<p<\mathrm{\infty }$, and from $L^1({\mathbb{R}}^n,{\mu }_{\nu })$ into $L^{1,\mathrm{\infty }}({\mathbb{R}}^n,{\mu }_{\nu })$. In addition, we obtain boundedness properties for the operator $G_{\nu ,M}^{k,\mathrm{\ell }}$, $1\le \mathrm{\ell }<2M$, defined by $$G_{\nu ,M}^{k,\mathrm{\ell }}(f)={\left({\int }_0^{\mathrm{\infty }}{\mid t^{\mathrm{\ell }\mathrm{/}2+k}{\mathrm{\partial }}_t^k{D}^{(\mathrm{\ell })}(I-t{\mathrm{\Delta }}_{\nu }{)}^{-M}(f)\mid }^2\frac{dt}{t}\right)}^{\frac{1}{2}},$$ for certain differentiation operator $D^{(\mathrm{\ell })}$.