$L^p$-boundedness properties for some harmonic analysis operators defined by resolvents for a Laplacian with drift in Euclidean spaces
Jorge J. Betancor, Juan C. Fariña, Lourdes Rodríguez-Mesa
Abstract
We consider the Laplacian with drift in $\mathbb R^n$ defined by $Δ_ν= \sum_{i=1}^n(\frac{\partial^2}{\partial x_i^2} + 2 ν_i\frac{\partial }{\partial{x_i}})$ where $ν=(ν_1,\ldots,ν_n)\in \mathbb R^n\setminus\{0\}$. The operator $Δ_ν$ is selfadjoint with respect to the measure $dμ_ν(x)=e^{2\langleν,x\rangle}dx$. 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 $$ W^k_{ν,M,*}(f) = \sup_{t>0}\left|A^k_{ν,M,t}(f)\right|,\hspace{5mm}g_{ν,M}^k(f) = \left(\int_0^\infty\left|A^k_{ν,M,t}(f)\right|^2\frac{dt}{t}\right)^{\frac{1}{2}},\,k\geq 1, $$ the $ρ$-variation operator $$ V_ρ\left( \{A^k_{ν,M,t}\}_{t>0}\right)(f)= \sup_{0<t_1<\cdots<t_\ell,\,\ell \in \mathbb N}\left(\sum^{\ell-1}_{j=1}\left|A^k_{ν,M,t_j}(f)- A^k_{ν,M,t_{j+1}}(f)\right|^ρ\right)^{\frac{1}ρ},\;\; ρ>2, $$ and, if $\{t_j\}_{j\in \mathbb N}$ is a decreasing sequence in $(0,\infty)$, the oscillation operator $$ O(\{A_{ν,M,t}^k\}_{t>0},\{t_j\}_{j\in \mathbb N})(f)=\Big(\sum_{j\in \mathbb N}\;\;\sup_{t_{j+1}\leq \varepsilon <\varepsilon '\leq t_j}|A^k_{ν,M,\varepsilon}(f)-A^k_{ν,M,\varepsilon '}(f)|^2 \Big)^{1/2}. $$ where $A^k_{ν,M,t}=t^k\partial^k_t(I-tΔ_ν)^{-M}$, $t>0$. We denote by $T_{ν,M}^k$ any of the above operators. We analyze the boundedness of $T^k_{ν,M}$ on $L^p(\mathbb R^n,μ_ν)$ into itself, for every $1<p<\infty$, and from $L^1(\mathbb R^n,μ_ν)$ into $L^{1,\infty}(\mathbb R^n,μ_ν)$. In addition, we obtain boundedness properties for the operator $G_{ν,M}^{k,\ell}$, $1\leq \ell <2M$, defined by $$ G_{ν,M}^{k,\ell}(f)=\left(\int_0^\infty\left|t^{\ell /2+k}\partial _t^kD^{(\ell)}(I-tΔ_ν)^{-M}(f) \right|^2\frac{dt}{t}\right)^{\frac{1}{2}}, $$ for certain differentiation operator $D^{(\ell)}$.