Abstract
Our goal is to obtain the John–Nirenberg inequality for ball Banach function spaces X, provided that the Hardy–Littlewood maximal operator M is bounded on the associate space X' by using the extrapolation. As an application we characterize BMO, the bounded mean oscillation, via the norm of X.
Highlights
The classical BMO semi-norm · BMO is defined by b BMO := sup Q:cube |Q|b(y) – mQ(b) dy = sup mQ QQ:cube b – mQ(b) for b ∈ L1loc(Rn)
Equivalent expressions of the BMO norm · BMO are necessary in order to prove boundedness of commutators involving BMO functions on various function spaces
Applying the inequality and the extrapolation again, we will give another proof of Theorem 1.2 in the setting of ball Banach function spaces
Summary
And below mQ(f ) denotes the average of the locally integrable function f over a cube Q. The BMO space consists of all locally integrable functions b such that b BMO < ∞. Due to the John–Nirenberg inequality and the L∞–BMO boundedness of singular integral operators, the BMO space is one of the important function spaces in real analysis. Equivalent expressions of the BMO norm · BMO are necessary in order to prove boundedness of commutators involving BMO functions on various function spaces. The estimate b BMO ≤ b BMOLp is obtained by the usual Hölder inequality. The opposite estimate C b BMOLp ≤ b BMO is not obvious. The following is a famous result named the John–Nirenberg inequality [22] which proves the estimate
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