Boundedness of Operators on Weighted Morrey–Campanato Spaces in the Bessel Setting

Let \(\lambda \in (-\frac{1}{2},\infty )\) $\lambda \in (-\frac{1}{2},\infty )$ , and \(\{\mathcal {W}_{t}^{\lambda }\}_{t>0}\) ${\{{\mathcal{W}}_t^{\lambda }\}}_{t>0}$ be the heat semigroup related to the Bessel Schrödinger operator \(S_{\lambda }:=-\frac{d^2}{dx^2}+\frac{\lambda ^2-\lambda }{x^2}\) $S_{\lambda }:=-\frac{d^2}{d{x}^2}+\frac{{\lambda }^2-\lambda }{x^2}$ on \(\mathbb {R}_{+}:=(0, \infty )\) ${\mathbb{R}}_{+}:=(0,\infty )$ . The authors introduce the weighted Morrey–Campanato space \(\mathrm{BMO^\alpha (\mathbb {R}_{+}, \omega )}\) ${\mathrm{BMO}}^{\alpha }({\mathbb{R}}_{+},\omega )$ with \(\alpha \in [0, 1)\) $\alpha \in [0,1)$ and \(\omega \in A_{\infty }(\mathbb {R}_{+})\) $\omega \in A_{\infty }({\mathbb{R}}_{+})$ , and show that for any weight function \(\omega \in RH_{s^{\prime }}(\mathbb {R}_{+})\cap A_{p/s}(\mathbb {R}_{+})\) $\omega \in R{H}_{s^{\prime }}({\mathbb{R}}_{+})\cap A_{p/s}({\mathbb{R}}_{+})$ , the oscillation, variation, radial maximal operator, and maximal operator of difference associated with the family \(\{t^m\partial _t^m\mathcal {W}_{t}^{\lambda }\}_{t>0}\) ${\{t^m{\partial }_t^m{\mathcal{W}}_t^{\lambda }\}}_{t>0}$ are bounded from \(\mathrm{BMO^\alpha (\mathbb {R}_{+}, \omega )}\) ${\mathrm{BMO}}^{\alpha }({\mathbb{R}}_{+},\omega )$ to its subspace \(\mathrm{BLO^\alpha (\mathbb {R}_{+}, \omega )}\) ${\mathrm{BLO}}^{\alpha }({\mathbb{R}}_{+},\omega )$ , where \(\lambda \in \mathbb {R}_{+}\) $\lambda \in {\mathbb{R}}_{+}$ , \(m\in {\mathbb {N}}\cup \{0\}\) $m\in \mathbb{N}\cup \{0\}$ , \(p\in (1,\infty )\) $p\in (1,\infty )$ , \(s\in [1,p)\) $s\in [1,p)$ such that \(p/s+\alpha <1+\min \{1,\lambda \}\) $p/s+\alpha <1+min\{1,\lambda \}$ , and \(s^{\prime }\) $s^{\prime }$ denotes the conjugate exponent of s. These results are new even in the case of \(\omega \equiv 1\) $\omega \equiv 1$ . As a corollary, the boundedness of these operators on spaces \(\mathrm{BMO^\alpha (\mathbb {R}_{+}, \omega )}\) ${\mathrm{BMO}}^{\alpha }({\mathbb{R}}_{+},\omega )$ is further established.