Professors Bellow and Jones will continue their ongoing investigation of almost-everywhere convergence questions arising in connection with the pointwise ergodic theorem along subsequences and along other sequences of operators. A.e. convergence in mean for certain arithmetic sequences will be an especial focus of interest. Calderon-Zygmund type decomposition techniques will be used to study such problems. The principal investigators will also pursue problems at the other extreme involving the strong sweeping out property, where a.e. convergence in the sup norm fails very badly. This project is mathematical research in ergodic theory. This theory is concerned with what happens on average over the long run when a suitable transformation of a suitable underlying space is iterated many times. (Typically, the underlying space might be just a line segment, and the transformation might cut the line segment up into pieces, then stretch or shrink and rearrange them.) One point of view is to study the behavior of functions on the space when their arguments are repeatedly subjected to the transformation. It frequently happens that the averages of the iterates of the functions converge in a meaningful fashion. A recent trend in the subject is to look at averages over subsequences, for instance the squares or the primes, instead of over all integer indices. Professors Bellow and Jones will pursue ergodic theory over subsequences in this sense.
