A reverse BMO-Hardy inequality
The major result proved in this note is the following: The Hardy operator defined by \[ (Pf)(x):= {1\over x} \int^x_0 f(t) dt,\quad x\in\mathbb{R}_+,\quad f\in L^1_{\text{loc}}(\mathbb{R}_+), \] is bounded on \(\text{BMO}(\mathbb{R}_+)\): \[ \|Pf\|_{\text{BMO}(\mathbb{R}_+)}\leq\|f\|_{\text{BMO}(\mathbb{R}_+)}.
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