In this paper we establish a variety or results in ergodic theory by using techniques from probability and analysis. We discuss divergence of operators, including strong sweeping out and Bourgain's entropy method. We consider square functions, oscillation operators, and variational operators for ergodic averages. We also consider almost everywhere convergence of convo-lution powers.

In Rn define (TXirf)~(£) = /(£)(! k_1í2l)+If n > 3, A > ¿(n-l)/(n+l)and2 and the associated maximal operators are r;/(x) = suP|(/-(i-Ki2)i)-|(x). r>0 It is conjectured that, when A > 0, T\ is bounded on Lp if and only if pó(A) < p < Po(A), where po(A)...

In the present paper we prove the a.e. convergence of Fejér means of integrable functions with respect to the two-dimensional representative product systems on a bounded compact totally disconnected group provided that the set of indices is in a cone-like set.

and Applied Analysis 3 Lemma 4. For λ > 0, one has 󵄩󵄩󵄩󵄩󵄩 K δ,l,j λ f (x) 󵄩󵄩󵄩󵄩󵄩 2 2 ≤ C2 −2M(j+l) δ 2M󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩 2 2 , (19) where the constant C is independent of λ and δ. Proof. With the method similar to the proof of Lemma 4 in [9], we write h(t) = φ(t) − φ(2t) and expandm into a Taylor series around λt. Then, ?̂? δ,l,j λ (t) = ∫m δ (λ(t − 2 −(j+l) δ 2 r λ )) ĥ (r) dr = ∫m δ (λt − 2 −(j+l) δ 2 ...

A preliminary version of this work appeared as [2]. y Supported in part by National Science Foundation grant CCR-9000045. Some of this research was performed while the author was a visiting professor at Institut f ur Informatik, Universit at W urzburg, D-8700 W urzburg, Federal Republic of Germany. z Supported in part by the National Science Foundation grants CCR-8808949 and CCR-8958528. Wo...

We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw be the lattice of degrees of unsolvability of mass proble...