Oscillation and variation for the Riesz transform associated with Bessel operators

Let lambda > 0 and let Delta(lambda) := - d(2)/dx(2) - 2 lambda/x d/dx be the Bessel operator on R+ := (0, infinity). We show that the oscillation operator O( R-Delta lambda,*) and variation operator V-rho(R-Delta lambda,*) of the Riesz transform R-Delta lambda associated with Delta(lambda) are both bounded on L-p(R+, dm(lambda)) for p is an element of(1, infinity), from L-1(R+, dm(lambda)) to L-1,L-infinity (R+, dm(lambda)), and from L-infinity(R+, dm(lambda)) to BMO(R+, dm(lambda)), where rho is an element of(2, infinity) and dm(lambda) (x) := x(2 lambda) dx. As an application, we give the corresponding L-p-estimates for beta-jump operators and the number of up-crossings.