Square roots of the Bessel operators and the related Littlewood–Paley estimates
Volume 263 / 2022
Studia Mathematica 263 (2022), 19-58
MSC: Primary 42B20; Secondary 42B25.
DOI: 10.4064/sm190922-19-11
Published online: 22 November 2021
Abstract
Let 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.
