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

Abstract

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