Variation operators for semigroups and Riesz transforms on \(BMO\) in the Schrödinger setting (Q1949218)

The Schrödinger operator on \(\mathbb{R}^n\), \(n\geq 3\), is defined by \[ \mathcal {L}=-\triangle+V, \] where the potential \(V\) is nonzero, nonnegative and belongs, for some \(q \geq n/2\), to the reverse Hölder class \(RH_q\), that is, there exists \(C>0\) such that, for every ball \(B\subset \mathbb{R}^n\), \[ \left(\frac{1}{|B|}\int_B V(x)^q dx\right)^{1/q}\leq C\frac{1}{|B|}\int_B V(x)dx. \] Suppose that \(\{T_t\}_{t>0}\) is a family of operators defined for functions in \(L^p(\mathbb{R}^n)\), for some \(1\leq p<\infty\), such that \(\lim_{t\rightarrow0^+}T_t f\) exists in \(L^p(\mathbb{R}^n)\). If \(\rho>2\), the variation operator \(V_{\rho}(T_t)\) is given by \[ V_{\rho}(T_t)(f)(x)=\sup_{{\{t_j\}_{j=0}^\infty}\downarrow 0}\left(\sum_{j=1}^\infty |T_{t_j}(f)(x)-T_{t_{j+1}}(f)(x)|^{\rho}\right)^{1/{\rho}}. \] Denote by \(\{W_t^\mathcal {L}\}_{t>0}\) the heat semigroup associated to \(\mathcal {L}\). For every \(t>0\), \[ W_t^\mathcal {L}(f)(x)=\int_{\mathbb{R}^n}W_t^\mathcal {L}(x,y) f(y) dy,\quad f\in L^2(\mathbb{R}^n). \] Let \(l=1,2,\ldots,n\). The Riesz transform \(R_{l}^{\mathcal{L}}\) can be extended to \(L^p(\mathbb{R}^n)\) as the principal value operator \[ R_{l}^{\mathcal{L}}(f)(x)=\lim_{\varepsilon\rightarrow 0^+} \int_{|x-y|>\varepsilon}R_{l}^{\mathcal{L}}(x,y) f(y) dy, \quad \text{a.e. \(x\in \mathbb{R}^n\),} \] where \[ R_{l}^{\mathcal{L}}(x,y)=-\frac{1}{2\pi}\frac{\partial}{\partial x_l}\int_R(-i\tau)^{-1/2}\Gamma(x,y,\tau) d\tau, \quad x,y\in \mathbb{R}^n, x\neq y, \] and \(\Gamma(x,y,\tau)\) represents the fundamental solution for the operator \(\mathcal{L}+i\tau\) [\textit{J. J. Betancor, J. C. Fariña, E. Harboure} and \textit{L. Rodríguez-Mesa}, ``\(L^p\)-boundedness properties of variation operators in the Schrödinger setting''. Rev. Mat. Complut. 26, 485--534 (2013)]. For every \(\varepsilon>0\), the \(\varepsilon\)-truncation \(R_{l}^{\mathcal{L},\varepsilon}\) of \(R_{l}^{\mathcal{L}}\) is defined as usual by \[ R_{l}^{\mathcal{L},\varepsilon}(f)(x)=\int_{|x-y|>\varepsilon}R_{l}^{\mathcal{L}}(x,y) f(y) dy, \quad l=1,2,\ldots,n. \] The authors obtain that the variation operators \(V_{\rho}(W_t^{\mathcal{L}})\) and \(V_{\rho}(R_{l}^{\mathcal{L},\varepsilon})\) are bounded from \(BMO^{\mathcal{L}}\) into itself. And they also yield analogous results in the classical settings.