A note on BMO and its application ✩

In this note, at first we will point out a fact which is implicitly contained in the original paper of John and Nirenberg [Comm. Pure Appl. Math. 14 (1961) 415–426]. If a BMO(Rn) function f satisfies lima→∞ |f |>a |f |dx <∞ (obviously if the value of left term is finite it must be zero), then there holds f ∗(t) Cf,n ln 2+ 1 t , (1) where f ∗(t) is the nondecreasing rearrangement of f . As a consequence, we get the following inequality f p Cnp f r/p r f 1−r/p BMO , 1 r p <∞. (2) By this inequality, the bilinear estimates in BMO in [Math. Z. 235 (2000) 173–194] can be easily improved. These bilinear estimates are crucial for the main results in that paper.  2004 Elsevier Inc. All rights reserved