Oscillation and variation for Riesz transform in setting of Bessel operators on H 1 and BMO
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Oscillation and variation for Riesz transform in setting of Bessel operators on H 1 and BMO Xiaona CUI1 , Jing ZHANG2 1 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China 2 School of Mathematics and Statistics, Yili Normal University, Yining 835000, China
c Higher Education Press 2020
2
d 2λ d Abstract Let λ > 0, and let the Bessel operator ∆λ := − dx 2 − x dx defined on R+ := (0, ∞). We show that the oscillation and ρ-variation operators of the Riesz transform R∆λ associated with ∆λ are bounded on BMO(R+ , dmλ ), where ρ > 2 and dmλ = x2λ dx. Moreover, we construct a (1, ∞)∆λ -atom as a counterexample to show that the oscillation and ρ-variation operators of R∆λ are not bounded from H 1 (R+ , dmλ ) to L1 (R+ , dmλ ). Finally, we prove that the oscillation and the ρ-variation operators for the smooth truncations associated e∆ are bounded from H 1 (R+ , dmλ ) to L1 (R+ , dmλ ). with Bessel operators R λ
Keywords Oscillation operator, variation operator, Bessel operator MSC 42B20, 42B25, 42B30, 42B35 1
Introduction
Let λ > 0 be a positive constant, and let ∆λ be the Bessel operator, which is defined by setting, for all suitable functions f on R+ := (0, ∞), ∆λ f (x) = −
d2 2λ d f (x) − f (x). dx2 x dx
As a pioneer study, Muckenhoup and Stein [20] built up a theory associated with ∆λ , which is similar to the classic one associated with the Laplace operator. Since then, a lot of work concerning the Bessel operator were considered, such as properties of the Riesz transform R∆λ f := ∂x (∆λ )−1/2 f, see, for example, [3,6,12,20,21,24]. In particular, Betancor et al. [6] deduced Received June 21, 2019; accepted July 13, 2020 Corresponding author: Xiaona CUI, E-mail: [email protected]
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Xiaona CUI, Jing ZHANG
that if 1 6 p < ∞ and f ∈ Lp (R+ , dmλ ), then, for almost every x ∈ R+ , Z R∆λ (x, y)f (y)dmλ (y), R∆λ f (x) = lim R∆λ ,t f (x) := lim t→0+
t→0+
|x−y|>t
where dmλ (y) = y 2λ dy, and for any x, y ∈ R+ with x 6= y, Z 2λ π (x − y cos θ)(sin θ)2λ−1 R∆λ (x, y) = − dθ. π 0 (x2 + y 2 − 2xy cos θ)λ+1
(1)
Moreover, the atomic Hardy space H 1 (R+ , dmλ ) associated to ∆λ in terms of the Riesz transform and the radial maximal function associated with the Hankel convolution of a class of suitable functions were characterized by Betancor et al. [5]. In this paper, we will focus on the boundedness investigation of the oscillation and variation operators for the Riesz transform in the setting of the Bessel operators on H 1 and BMO. We first recall some notations and backgrounds. Let T∗ = {Tt }t>0 be a family of bounded operators such that the limit limt→0+ Tt f exists in some sense. A classical method to measure the speed of the convergence of {Tt }t>0 is to consider the square functions of the type X ∞
|Ttj f − Ttj+1 f |
2
1/2
j=1
for any given sequence {tj }j , which decreases to 0. Recently, it is interesting to consider the oscillation and variation operators of T∗ . To be precise, for each fixed sequence {tj }j in R+ which decreases to 0, the oscillation operator
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