Square roots of the Bessel operators and the related Littlewood–Paley estimates
Volume 263 / 2022
Abstract
Let $\Delta _{\lambda }$ and $S_{\lambda }$, $\lambda \in \mathbb {R}_+:=(0,+\infty )$, be the two Bessel operators studied by Muckenhoupt–Stein (1965). We prove that the square root of the Bessel operators and the corresponding “gradient” operators are equivalent in $L^p$ spaces for $1 \lt p \lt \infty $. Moreover, by using holomorphic functional calculus, we establish the characterizations of boundedness on $L^p$ spaces associated with Bessel operators in terms of the Littlewood–Paley $g$-function with respect to the square root of the Bessel operator. Also, we study boundedness properties of Littlewood–Paley $g$-function associated with the square root of the Bessel operator on the odd $\rm {BMO}$ space $\rm {BMO}_+$ and the atomic Hardy space $H^1$.
