Almost everywhere convergence of powers of some positive L-p contractions

作者：Cohen Guy^*; Cuny Christophe; Lin Michael

来源：Journal of Mathematical Analysis and Applications, 2014, 420(2): 1129-1153.

DOI：10.1016/j.jmaa.2014.06.014

• Ritt operators
• Positive contractions
• Almost everywhere convergence
• Products of conditional expectations
• Convolution powers of group actions
• Fractional coboundaries

We extend the solution of Burkholder%26apos;s conjecture for products of conditional expectations, obtained by Delyon and Delyon for L-2 and by Cohen for L-p, 1 %26lt; p %26lt; infinity, to the context of Badea and Lyubich: Let T be a finite convex combination of operators T-j which are products of finitely many conditional expectations. Then T-n f converges a.e. for every f is an element of L-p, 1 %26lt; p %26lt; infinity, with sup(n) vertical bar T-n f vertical bar is an element of L-p. The proof uses the work of Le Merdy and Xu on positive L-p contractions satisfying Ritt%26apos;s resolvent condition. As another application of the work of Le Merdy and Xu, we extend a result of Bellow, Jones and Rosenblatt, proving that if a probability {a(k)}(k is an element of z) has bounded angular ratio, then for every positive invertible isometry S of an L-p space (1 %26lt; p %26lt; infinity), the operator T = Sigma(k is an element of z) a(k)S(k) is a positive L-p contraction such that for every f is an element of L-p, Tn f converges a.e. and sup(n) vertical bar T-n f vertical bar is an element of L-p. If {a(k)} is supported on N, the same result is true when S is only a positive contraction of L-p. Similar results are obtained for a-averages of bounded continuous representations of a a-compact LCA group by positive operators in one L-p space, 1 %26lt; p %26lt; infinity. For a positive contraction T on L-p which satisfies Rift%26apos;s condition and f is an element of (I - T)(alpha) L-p (0 %26lt; alpha %26lt; 1) we prove that n(alpha)T(n) f -%26gt; 0 a.e., and sup(n) n(alpha) vertical bar T-n f vertical bar is an element of L-p.

• 出版日期2014-12-15