Let $\lambda \in (-\frac{1}{2},\infty )$ and $S_{\lambda }≔-\frac{d^2}{d{x}^2}+\frac{{\lambda }^2-\lambda }{x^2}$ be the Bessel Schrödinger operator on ${\mathbb{R}}_{+}≔(0,\infty )$. The authors obtain the sharp power-weighted $L^p$, weak type and restricted weak type inequalities for the oscillation operator ${\mathcal{O}}_{{\left\{t_i\right\}}_{i\in \mathbb{N}}}({\left\{t^m{\partial }_t^m{\mathcal{W}}_t^{\lambda }\right\}}_{t>0},\cdot )$ and the variation operator ${\mathcal{V}}_{\rho }({\left\{t^m{\partial }_t^m{\mathcal{W}}_t^{\lambda }\right\}}_{t>0},\cdot )$ of the heat semigroup ${\left\{{\mathcal{W}}_t^{\lambda }\right\}}_{t>0}$ associated with $S_{\lambda }$, where $\rho \in (2,\infty )$ and $m\in {\mathbb{Z}}_{+}≔\mathbb{N}\cup \left\{0\right\}$. Moreover, for $\lambda \in (0,\infty )$, the boundedness of ${\mathcal{O}}_{{\left\{t_i\right\}}_{i\in \mathbb{N}}}({\left\{t^m{\partial }_t^m{\mathcal{W}}_t^{\lambda }\right\}}_{t>0},\cdot )$ and ${\mathcal{V}}_{\rho }({\left\{t^m{\partial }_t^m{\mathcal{W}}_t^{\lambda }\right\}}_{t>0},\cdot )$ from the Hardy space $H^p\left({\mathbb{R}}_{+}\right)$ into $L^p\left({\mathbb{R}}_{+}\right)$ with $p\in (\frac{1}{2},1]$ and on the Campanato type spaces BMO${}^{\alpha }$ $\left({\mathbb{R}}_{+}\right)$ with $\alpha \in [0,1)\cap (0,\lambda )$ are obtained.

让 $\lambda \in （-\frac{\mathrm{1个}}{2}，\infty ）$ 和 ${\mathrm{小号}}_{\lambda }≔-\frac{d^2}{d{X}^2}+\frac{{\lambda }^2-\lambda }{X^2}$ 成为BesselSchrödinger的运营商 ${\mathbb{[R}}_{+}≔（0，\infty ）$。作者获得了敏锐的功率加权${\mathrm{大号}}^p$振荡算子的，弱型和受限弱型不等式 ${\mathcal{Ø}}_{{\left\{{Ť}_{\mathrm{一世}}\right\}}_{\mathrm{一世}\in \mathbb{ñ}}}（{\left\{{Ť}^{米}{\partial }_{Ť}^{米}{\mathcal{w\; ^}}_{Ť}^{\lambda }\right\}}_{Ť>0}，\cdot ）$ 和变异算子 ${\mathcal{V}}_{\rho }（{\left\{{Ť}^{米}{\partial }_{Ť}^{米}{\mathcal{w\; ^}}_{Ť}^{\lambda }\right\}}_{Ť>0}，\cdot ）$ 热半群的 ${\left\{{\mathcal{w\; ^}}_{Ť}^{\lambda }\right\}}_{Ť>0}$ 有关联 ${\mathrm{小号}}_{\lambda }$，在哪里 $\rho \in （2，\infty ）$ 和 $米\in {\mathbb{ž}}_{+}≔\mathbb{ñ}\cup \left\{0\right\}$。而且，对于$\lambda \in （0，\infty ）$， ${\mathcal{Ø}}_{{\left\{{Ť}_{\mathrm{一世}}\right\}}_{\mathrm{一世}\in \mathbb{ñ}}}（{\left\{{Ť}^{米}{\partial }_{Ť}^{米}{\mathcal{w\; ^}}_{Ť}^{\lambda }\right\}}_{Ť>0}，\cdot ）$ 和 ${\mathcal{V}}_{\rho }（{\left\{{Ť}^{米}{\partial }_{Ť}^{米}{\mathcal{w\; ^}}_{Ť}^{\lambda }\right\}}_{Ť>0}，\cdot ）$ 来自Hardy空间 $H^p（{\mathbb{[R}}_{+}）$ 进入 ${\mathrm{大号}}^p（{\mathbb{[R}}_{+}）$ 与 $p\in （\frac{\mathrm{1个}}{2}，\mathrm{1个}]$ 在Campanato型空间BMO上${}^{\alpha }$ $（{\mathbb{[R}}_{+}）$ 与 $\alpha \in [0，\mathrm{1个}）\cap （0，\lambda ）$ 获得。