In $\mathbb{R}^n$, suppose $f \in \mathrm{BMO}$ and $\phi \in \mathrm{C}_0^\infty$, then can we show that the convolution $f * \phi \in \mathrm{BMO}$?
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Yes, we can. See for example: Harmonic Analysis by Stein. Chapter 4, section 4. – Pocho la pantera Sep 28 '13 at 03:10