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BMO and commutators of martingale transforms

Svante Janson — 1981

Annales de l'institut Fourier

The commutator of multiplication by a function and a martingale transform of a certain type is a bounded operator on L p , 1 < p < , if and only if the function belongs to BMO. This is a martingale version of a result by Coifman, Rochberg and Weiss.

Branching processes, and random-cluster measures on trees

Geoffrey GrimmettSvante Janson — 2005

Journal of the European Mathematical Society

Random-cluster measures on infinite regular trees are studied in conjunction with a general type of ‘boundary condition’, namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree T of a branching process. What is the...

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