Oscillation and Variation for the Hilbert Transform

Hf (x)= lim →0+ H f (x). It is well known that this limit exists a.e. for all f ∈ L, 1 ≤ p < ∞. In this paper, we will consider the oscillation and variation of this family of operators as goes to zero, which gives extra information on their convergence as well as an estimate on the number of λ-jumps they can have. For earlier results on oscillation and variation operators in analysis and ergodic theory, including some historical remarks and applications, the reader may look in [2], [3], [5], [4], and [6]. For each fixed sequence (ti)↘ 0, we define the oscillation operator ( H∗f ) (x)= ( ∞ ∑