The Convergence of Lebesgue Derivatives and Ergodic Averages
We study certain operators defined by infinite series that describe the nature of convergence of stochastic processes; these include square functions, oscillation operators, and variation operators. The goal is to prove that these operators map Linfinity to BMO and are of strong type (p, p) where 1 < p < infinity for the case that the stochastic processes are Lebesgue differentiation or ergodic averages. In Chapter 2, we prove the appropriate sublinear operator interpolation between the weak type (1, 1) estimate and the strong estimate from Linfinity to BMO. In Chapter 3, we prove that these operators map Linfinity to BMO and are of strong type (p, p) which 1 < p < infinity for Lebesgue derivatives. In Chapter 4, we prove that these operators for ergodic averages are of strong type (p, p) for 1 < p < infinity. In the last chapter, we characterize the strong estimate from Linfinity to L infinity for these operators. We also construct explicit counterexamples to show that the role of BMO is vital since these operators do not map Linfinity to L infinity in general.
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign