\(L^2(\mathbb{R}^n)\) boundedness for the commutators of convolution operators (Q2764668)

By localization and the Fourier transform estimates, the author obtains the following result:NEWLINENEWLINENEWLINELet \(K(x)\) be a function on \(\mathbb{R}^n\setminus\{0\}\) and \(K(x)= \sum_{j\in\mathbb{Z}} K_j(x)\). Let \(k\) be a positive integer. Suppose that there are some constants \(C>0\), \(0< A<1/2\) and \(\alpha>k+1\) such that for each \(j\in\mathbb{Z}\) NEWLINE\[NEWLINE\|K_j\|_1\leq C,\quad\|\nabla\widehat K_j\|_\infty\leq C 2^j,NEWLINE\]NEWLINE NEWLINE\[NEWLINE|\widehat K_j(\xi)|\leq C\min\{A|2^j\xi|, \log^{-\alpha}(2+|2^j\xi|)\}.NEWLINE\]NEWLINE Then for \(b\in \text{BMO}(\mathbb{R}^n)\) and \(0<\nu<1\), such that \(\alpha\nu> k+1\), the commutator NEWLINE\[NEWLINET_{b,k} f(x)= \int_{\mathbb{R}^n} (b(x)- b(y))^k K(x-y) f(y) dy,\quad f\in C^\infty_0(\mathbb{R}^n)NEWLINE\]NEWLINE is bounded on \(L^2(\mathbb{R}^n)\) with bound \(C(n,k,\alpha,\nu)\log^{- \alpha\nu+ k+1}(1/A)\|b\|^k_{\text{BMO}}\).NEWLINENEWLINENEWLINEAs applications, the author considers the special cases of \(T_{b,k}\): NEWLINE\[NEWLINE\widetilde T_{b,k}f(x)= \int_{\mathbb{R}^n} (b(x)- b(y))^k \Omega(x- y)|x-y|^{- n} f(y) dy,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\overline T_{b,k}f(x)= \int_{\mathbb{R}^n}(b(x)- b(y))^k h(|x-y|) \Omega(x-y)|x-y|^{- n} f(y) dy,NEWLINE\]NEWLINE where \(\Omega\) is a homogeneous function of degree zero, integrable on the unit sphere \(S^{n-1}\), and it has mean value zero on \(S^{n-1}\); \(h\) is a measurable function satisfying \(\sup_{R>0} \int^{2R}_R|h(r)|^s r^{-1} dr< \infty\) for some \(s>1\). Under certain size conditions on \(\Omega\), the author obtains \(L^2\) boundedness for these two operators. These results extend some known theorems.