\(L^p(\mathbf R^n)\) boundedness for the commutators of convolution operators (Q2755695)

Let \(B_j=B(0,2^j)\) be the open ball centered at the origin with radius \(2^j\) and let \(K\) be a function defined on \({\mathbb{R}}^n\setminus \{0\}\) satisfying the following properties: for \(j\in {\mathbb{Z}}\), NEWLINE\[NEWLINE\begin{aligned} & \|K_j\|_{L(\log L)^k, B_{j+1}\setminus B_j}\leq C2^{-jn},\\ &|\widehat {K_j}(\xi)|\leq C\min \{2^{j}\xi|,\;\log^{-\alpha}(2+|2^j\xi|\},\\ & \|\bigtriangledown \widehat {K_j}\|_{L^\infty}\leq C2^j.\end{aligned}NEWLINE\]NEWLINE Here NEWLINE\[NEWLINE K_j(x)=K(x)\chi_{2^2\leq |x|<2^{j+1}}(x)=K(x)\chi_{B_{j+1}\setminus B_j}(x). NEWLINE\]NEWLINE In this paper, the authors prove that for any natural integer \(k\) and \(b\in \text{BMO}({\mathbb{R}}^n)\) and \({2\alpha\over 2\alpha-k-1}<p< {2\alpha\over k+1}\), the commutator operator NEWLINE\[NEWLINE T_{b,k}(f)(x)=\int_{{\mathbb{R}}^n}(b(x)-b(y))^k K(x-y)f(y)dy,.\qquad f\in C^\infty_0({\mathbb{R}}^n) NEWLINE\]NEWLINE is bounded on \(L^p({\mathbb{R}}^n)\) and the operator norm is bounded by \(C(n,k,p,\alpha)\|b\|_{\text{BMO}}^k\). The authors also obtain an application of the above result to weighted commutator operators.