Boundedness of multilinear operators with rough kernel on Herz spaces (Q1423954)

The authors treat the multilinear operator defined by \[ T_{A,\Omega}f(x)=T(| x-\cdot| ^{-m+1}R_{m}(A;x,\cdot)f(\cdot))(x), \] where \(\Omega\) is a homogeneous function of degree zero, \(T\) is a linear operator satisfying \[ | Tf(x)| \leq C| x| ^{-n}\int_{\mathbb R^n}| \Omega(x-y)f(y)| \,dy, \tag{a} \] for any \(f\in L^1\) with \(\text{supp}\,f\subset \{2^{k-1}<| x| \leq 2^k\}\) and \(| x| \geq 2^{k+1}\) \((k\in \mathbb Z)\), and \[ | Tf(x)| \leq C2^{-kn}\int_{\mathbb R^n}| \Omega(x-y)f(y)| \,dy, \tag{b} \] for any \(f\in L^1\) with \(\text{supp}\,f\subset \{2^{k-1}<| x| \leq 2^k\}\) and \(| x| \leq 2^{k-2}\) \((k\in \mathbb Z)\). Here, \[ R_m(A;x,y)=A(x)-\sum_{| \beta| <m}\frac{1}{\beta!} D^\beta A(y)(x-y)^\beta. \] They give: Let \(m\) be a positive integer, \(D^\beta A \in \text{BMO}\) for \(| \beta| =m-1\), \(\Omega\in L^r(S^{n-1})\), and suppose \(T_{A,\Omega}\) is bounded on \(L^q(\mathbb R^n)\) for some \(1<q<\infty\). Then, \(T_{A,\Omega}\) is bounded on the homogeneous Herz space \(\dot K_q^{\alpha,p}(\mathbb R^n)\) if \(\alpha,q\) satisfy one of the following conditions: (i) \(r'<q\) and \(-n/q<\alpha<n(1-1/q-(n-1)/(nr))\); (ii) \(r>q\) and \(n(1/r-1/q)-1/r<\alpha<n(1-1/q)\). This corresponds to the multilinear operator of singular integrals. They consider also multilinear operators corresponding to fractional integral operators. These multilinear operators are commutators in the case \(m=1\). So, their results extend corresponding ones in [\textit{S. Lu}, \textit{L. Tang} and \textit{D. Yang}, Sci. China, Ser. A 41, No. 10, 1023--1033 (1998; Zbl 0929.42009)] and [\textit{L. Liu}, Beijing Shifan Daxue Xuebao, Natural Sci. 38, 290--302 (2002)].