Multilinear commutators of Calderón-Zygmund operator on generalized variable exponent Morrey spaces (Q2055401)

Let \(\mathfrak{b}=(b_{1},\dots,b_{m})\) and \(b_{j}\) for \(1\leq j\leq m\), be locally integrable functions. Then multilinear commutators of Calderón-Zygmund operators are defined by \[ T_{\mathfrak{b}}f(x)=\int_{\mathbb{R}^{n}}\prod_{j=1}^{m}(b_{j}(x)-b_{j}(y))K(x,y)f(y)dy, \] where \(K(x,y)\) is Calderón-Zygmund kernel and \[ Tf(x)=\int_{\mathbb{R}^{n}}K(x,y)f(y)dy \] is the Calderón-Zygmund singular integral operator. We notice that when \(m=1\), \(T_{\mathfrak{b}}\) is the classical commutator which was introduced by Coifman, Rochberg, and Weiss [\textit{R. R. Coifman} et al., Ann. Math. (2) 103, 611--635 (1976; Zbl 0326.32011)]. The main results of the paper are the following: \par (1) Let \(p\in \mathcal{LH}(\mathbb{R}^{n})\cap P(\mathbb{R}^{n})\), and \((\phi_{1},\phi_{2})\) satisfy the condition \[ \int_{r}^{\infty}\ln^{m}\left(e+\frac{t}{r}\right) \frac{\operatorname{essinf}_{t<s<\infty}\phi_{1}(x,s)s^{\theta_{p}(x,s)}}{t^{\theta p(x,t)}} \frac{dt}{t}\leq C\phi_{2}(x,r), \] where \(C\) does not depend on \(x\) and \(r\). Let also \(\mathfrak{b} =(b_{1},\dots,b_{m})\) \(\in BMO^{m}(\mathbb{R}^{n})\). Then \(T_{\mathfrak{b}}\) is a bounded operator from \(M^{p(\cdot),\phi_{1}}\) to \(M^{p(\cdot),\phi_{2}},\) where \(\mathcal{LH}(\mathbb{R}^{n})\) is the class of globally log-Hölder continuous functions and \(M^{p(\cdot),\phi}\equiv M^{p(\cdot),\phi}(\mathbb{R}^{n})\) is the generalized variable exponent Morrey space. Moreover, it holds that \[ \left\Vert T_{\mathcal{b}}f\right\Vert_{M^{p(\cdot),\phi_{2}}}\lesssim \prod_{j=1}^{m}\left\Vert b_{j}\right\Vert_{BMO}\left\Vert f\right\Vert_{M^{p(\cdot),\phi_{1}}} \] \par (2) Let \(\mathfrak{b}=(b_{1},\dots,b_{m})\) \(\in BMO^{m}(\mathbb{R}^{n})\) , and \((\phi_{1},\phi_{2})\) satisfy the condition \[ \int_{r}^{\infty}\ln^{m}\left(e+\frac{t}{r}\right) \frac{\operatorname{essinf}_{t<s<\infty}\phi_{1}(x,s)s^{n}}{t^{n}}\frac{dt}{t}\leq C\phi_{2}(x,r), \] where \(C\) does not depend on \(x\) and \(r\). Let also \(\mathfrak{b} =(b_{1},\dots,b_{m})\) \(\in BMO^{m}(\mathbb{R}^{n})\). Then \(T_{\mathfrak{b}}\) is a bounded operator from \(M^{\Phi ,\phi_{1}}\) to \(WM^{1,\phi_{2}}\), where \(\Phi (t)=t\ln^{m}(e+t)\). Moreover, it holds that \[ \left\Vert T_{\mathfrak{b}}f\right\Vert_{WM^{1,\phi_{2}}}\lesssim \prod_{j=1}^{m}\left\Vert b_{j}\right\Vert_{BMO}\left\Vert f\right\Vert_{M^{\Phi ,\phi_{1}}} \]