Generalized higher commutators generated by the multilinear fractional integrals and Lipschitz functions (Q2925619)

Let \(\ell \in \mathbb N \setminus \{0\}\) and \(x_i \in \mathbb R^n\) for \(i=1, 2, \cdots, \ell\). Assume that \(m \in \mathbb N\) and \(F\) is a function with derivatives of order \(m\) on \(\mathbb R^n\), and denote NEWLINE\[NEWLINEL_{m+1}(F;x,y) = F(x) - \Sigma_{|\alpha| \leq m} \, \frac{1}{\alpha !}\, D^{\alpha} F(y)(x-y)^{\alpha}. NEWLINE\]NEWLINE Let \(\vec{F}=(F_1, \cdots, F_{\ell})\) be a finite collection of functions with derivative of order \(m\) on \(\mathbb R^n\) and \(f_i \in L^{\infty}_{c}(\mathbb R^n)\) for \(i=1, 2, \cdots, \ell\). For \(x \notin \cap_{i=1}^{\ell} \text{supp} f_i\), the \(\ell\) th generalized higher commutators generated by the multilinear fractional integral are defined by NEWLINE\[NEWLINE T_{s, \ell}^{\vec{F}}(\vec{f})(x) = \int_{ (\mathbb R^n)^{\ell}} \frac{\prod_{i=1}^{\ell}L_{m+1}(F_i; x, y_i)f_i(y_i)}{|(x-y_1, \cdots, x-y_{\ell}) |^{\ell n + (m_1 +\cdots+ m_{\ell})-s} } \;dy_1 \cdots dy_{\ell}, NEWLINE\]NEWLINE where \(0< s <\ell n + (m_1 +\cdots+ m_{\ell})\) and \(|(x_1, x_2, \cdots, x_{\ell})|\) is the norm of \((x_1, x_2, \cdots, x_{\ell}) \in (\mathbb R^n)^{\ell}\). The authors prove that the commutators \(T_{s, \ell}^{\vec{F}}\) are bounded from the product Lebesgue spaces to Lipschitz spaces and Triebel-Lizorkin spaces under suitable conditions.