Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ define$$[b,T](f)=b\cdot Tf-T(bf).$$ Question. Is $[b,T]$ is well-defined on $C_{0}^{\infty}(\mathbb{R}^{n})$ for all $b\in BMO(\mathbb{R}^{n})$?
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$\begingroup$ Do you want pointwise convergence of the defining integral? Or is it enough that $[b,T]$ sends $L^p$ into $L^p$ or similar? $\endgroup$– Eric ThomaApr 9, 2016 at 22:43
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$\begingroup$ Note $[b,T]$ is bounded as an operator $L^p \to L^p$ for $1 < p < \infty$. By the John-Nirenberg inequality e.g., the function $bf$ lies in any $L^p$, $p < \infty$. Assuming some smoothness on $K$, the integral for $Tf$ converges pointwise a.e. for $f \in L^p$. I am not sure of the necessity of smoothness though. $\endgroup$– Eric ThomaApr 9, 2016 at 22:47
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