$L^p$-boundedness properties for some harmonic analysis operators defined by resolvents for a Laplacian with drift in Euclidean spaces

Abstract

We consider the Laplacian with drift in ${\mathbb{R}}^n$ defined by ${\Delta }_{\nu }={\sum }_{i=1}^n(\frac{{\partial }^2}{\partial x_i^2}+2{\nu }_i\frac{\partial }{\partial x_i})$ where $\nu =({\nu }_1,\dots ,{\nu }_n)\in {\mathbb{R}}^n\setminus \{0\}$. The operator ${\Delta }_{\nu }$ 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 $$W_{\nu ,M,\ast }^k(f)=\underset{t>0}{\sup }\left|A_{\nu ,M,t}^k(f)\right|,g_{\nu ,M}^k(f)={\left({\int }_0^{\infty }{\left|A_{\nu ,M,t}^k(f)\right|}^2\frac{dt}{t}\right)}^{\frac{1}{2}},k\ge 1,$$ the $\rho$-variation operator $$V_{\rho }\left(\{A_{\nu ,M,t}^k{\}}_{t>0}\right)(f)=\underset{0<t_1<\cdots <t_{\ell },\ell \in \mathbb{N}}{\sup }{\left(\sum _{j=1}^{\ell -1}{\left|A_{\nu ,M,t_j}^k(f)-A_{\nu ,M,t_{j+1}}^k(f)\right|}^{\rho }\right)}^{\frac{1}{\rho }},\rho >2,$$ and, if $\{t_j{\}}_{j\in \mathbb{N}}$ is a decreasing sequence in $(0,\infty )$, the oscillation operator $$O(\{A_{\nu ,M,t}^k{\}}_{t>0},\{t_j{\}}_{j\in \mathbb{N}})(f)=(\sum _{j\in \mathbb{N}}\underset{t_{j+1}\le \epsilon <{\epsilon }^{\prime }\le t_j}{\sup }|A_{\nu ,M,\epsilon }^k(f)-A_{\nu ,M,{\epsilon }^{\prime }}^k(f){|}^2{)}^{1/2}.$$ where $A_{\nu ,M,t}^k=t^k{\partial }_t^k(I-t{\Delta }_{\nu }{)}^{-M}$, $t>0$. 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 $1<p<\infty$, and from $L^1({\mathbb{R}}^n,{\mu }_{\nu })$ into $L^{1,\infty }({\mathbb{R}}^n,{\mu }_{\nu })$. In addition, we obtain boundedness properties for the operator $G_{\nu ,M}^{k,\ell }$, $1\le \ell <2M$, defined by $$G_{\nu ,M}^{k,\ell }(f)={\left({\int }_0^{\infty }{\left|t^{\ell /2+k}{\partial }_t^k{D}^{(\ell )}(I-t{\Delta }_{\nu }{)}^{-M}(f)\right|}^2\frac{dt}{t}\right)}^{\frac{1}{2}},$$ for certain differentiation operator $D^{(\ell )}$.