A note on BMO and its application

Abstract: 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 function f satisfies (obviously if the value of left term is finite it must be zero), then there holds where is the nondecreasing rearrangement of f. As a consequence, we get the following inequality 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. [Copyright &y& Elsevier]