Abstract
This chapter focuses on an inequality of Kahane. It presents an assumption where (X, ∥·∥) is a Banache space and x(t) is a function from [0,1] to X. As the Rademacher functions are independent and identically distributed, ∥∑k≥l rk−l (t)xk∥ and ∥∑k≥l rk(t)xk∥ are identically distributed functions of t. The well-known theorem of John-Nirenberg concerning the distribution function of functions in BMO extends without change to vector valued BMOd.
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