Abstract
The principal result of this paper is that the convex combination of two positive, invertible, commuting isometries ofL p(X,F, μ) 1<p<+∞, one of which is periodic, admits a dominated estimate with constantp/p−1. In establishing this, the following analogue of Linderholm’s theorem is obtained: Let σ and ε be two commuting non-singular point transformations of a Lebesgue Space with τ periodic. Then given ε>O, there exists a periodic non-singular point transformation σ′ such that σ′ commutes with τ and μ(x:σ′x≠σx}<ε. Byan approximation argument, the principal result is applied to the convex combination of two isometries ofL p (0, 1) induced by point transformations of the form τx=x k,k>0 to show that such convex combinations admit a dominated estimate with constantp/p−1.
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Research supported in part by NSF Grant No. GP-7475. A portion of the contents of this paper is based on the author’s doctoral dissertation written under the direction of Professor R. V. Chacon of the University of Minnesota.
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Olsen, J. Dominated estimates of convex combinations of commuting isometries. Israel J. Math. 11, 1–13 (1972). https://doi.org/10.1007/BF02761444
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DOI: https://doi.org/10.1007/BF02761444