Variation and oscillation operators associated to semigroup generated by Schrödinger operator with fractional power

In this paper, the focus will be on the boundedness of variation operators, oscillation operators and λ-jump operators associated to semigroup generated by Schrödinger operator with Hardy potential L_a,α=(−△)^{[Formula presented]}+a|x|^−α,α∈(0,min⁡{2,n}). For a≥0, we will show the boundedness of the variation and oscillation operators associated to semigroup {e^−tL_a,α}_t>0 on L^p(Rⁿ,w) for every 1<p<∞ and w∈A_p(Rⁿ), and from L¹(Rⁿ) into L^1,∞(Rⁿ). In addition, we will show their boundedness from H_La,α^p(Rⁿ), the Hardy spaces associated to L_a,α, into L^p(Rⁿ) for [Formula presented]<p≤1. Furthermore, for a^⁎<a<0, we will show the boundedness of these operators on L^p(Rⁿ,w) for [Formula presented]<p<[Formula presented] and A_{[Formula presented]}(Rⁿ)∩RH_{([Formula presented])^′}(Rⁿ), where a^⁎ and σ are defined in the introduction of this paper.