$L^p$-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 $\nu=(\nu_1,\ldots,\nu_n)\in \mathbb R^n\setminus\{0\}$. The operator $\Delta_\nu$ is selfadjoint with respect to the measure $d\mu_\nu(x)=e^{2\langle\nu,x\rangle}dx$. 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 $$ W^k_{\nu,M,*}(f) = \sup_{t>0}\left|A^k_{\nu,M,t}(f)\right|,\hspace{5mm}g_{\nu,M}^k(f) = \left(\int_0^\infty\left|A^k_{\nu,M,t}(f)\right|^2\frac{dt}{t}\right)^{\frac{1}{2}},\,k\geq 1, $$ the $\rho$-variation operator $$ V_\rho\left( \{A^k_{\nu,M,t}\}_{t>0}\right)(f)= \sup_{0<t_1<\cdots<t_\ell,\,\ell \in \mathbb N}\left(\sum^{\ell-1}_{j=1}\left|A^k_{\nu,M,t_j}(f)- A^k_{\nu,M,t_{j+1}}(f)\right|^\rho\right)^{\frac{1}{\rho}},\;\; \rho>2, $$ and, if $\{t_j\}_{j\in \mathbb N}$ is a decreasing sequence in $(0,\infty)$, the oscillation operator $$ O(\{A_{\nu,M,t}^k\}_{t>0},\{t_j\}_{j\in \mathbb N})(f)=\Big(\sum_{j\in \mathbb N}\;\;\sup_{t_{j+1}\leq \varepsilon <\varepsilon '\leq t_j}|A^k_{\nu,M,\varepsilon}(f)-A^k_{\nu,M,\varepsilon '}(f)|^2 \Big)^{1/2}. $$ where $A^k_{\nu,M,t}=t^k\partial^k_t(I-t\Delta_\nu)^{-M}$, $t>0$. We denote by $T_{\nu,M}^k$ any of the above operators. We analyze the boundedness of $T^k_{\nu,M}$ 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 $G_{\nu,M}^{k,\ell}$, $1\leq \ell <2M$, defined by $$ G_{\nu,M}^{k,\ell}(f)=\left(\int_0^\infty\left|t^{\ell /2+k}\partial _t^kD^{(\ell)}(I-t\Delta _\nu)^{-M}(f) \right|^2\frac{dt}{t}\right)^{\frac{1}{2}}, $$ for certain differentiation operator $D^{(\ell)}$.