A multidimensional singular integral operator in the spaces defined by conditions on the \(k\)th order mean oscillation. (Q1595679)

Consider the singular integral operator \[ Af(x):= \int_{\mathbb R^ n}\Big( K(x-y)-\sum_{| \nu| \leq k-1} \frac1{\nu!}D^\nu K_*(-y)\,x^\nu\Big)\,f(y)\,dy, \] where \(k\in\mathbb N\), \(\nu=(\nu_1,\dots,\nu_ n)\in\mathbb N_0^ n\), \(\nu!:=\nu_1!\dots\nu_ n!\), \(D^\nu:=\frac{\partial^{\nu_1+\dots+\nu_ n}} {\partial x_1^{\nu_1} \dots\partial x_ n^{\nu_ n}}\), \(x^\nu:=x_1^{\nu_1}\dots x_ n^{\nu_ n}\), \(K(x):=\omega(x/| x| )| x| ^{-n}\) with \(\int_{S^{n-1}}\omega(x)\,dS=0\) (\(S^{n-1}\) is the unit sphere in \(\mathbb R^ n\)), assuming, for \(k=1\), that \(K(x)\) is differentiable and its first-order partial derivatives are bounded, and, for \(k>1\), that \(K(x)\) is \(k\) times continuously differentiable on \(S^ n\), \(K_*(x):=g_*(x)K(x)\) where \(g_*(x)\) is a smooth cut-off function supported in the set \(\{x\in\mathbb R^ n; | x| \geq1\}\) and satisfying \(g_*(x)=1\) for all \(x\), \(| x| \geq2\). Moreover, consider the operator \[ Tf(x):= \int_{\mathbb R^ n}K(x-y)\,f(y)\,dy. \] The author derives boundedness of the operators \(A\) and \(T\) on various function spaces of BMO type. All the results are presented without proofs. For author's previous related results see [Sov. Math. Dokl. 42, No. 2, 520--523 (1991); translation from Doklady Akad. Nauk SSSR 314, No. 3, 562--565 (1990; Zbl 0735.47018)].