Commutators of \(n\)-dimensional rough Hardy operators (Q547330)

The authors analyze commutators between \(M_b(f)= bf\) and \(n\)-dimensional rough Hardy operators \[ \mathcal H_{\Omega,\beta}f(x)=\frac{1}{|x|^{n-\beta}}\int_{|t|<|x|}\Omega(x-t)f(t)\, dt \] for \(x\in \mathbb R^n \setminus \{0\}\), where \(\Omega\in L^s(S^{n-1})\), \(1\leq s<\infty\), is homogeneous of degree zero and \(\beta\in \mathbb R\) acting on Lebesgue, Herz and central Morrey spaces. These commutators are denoted \(\mathcal H^b_{\Omega,\beta}= [b,\mathcal H_{\Omega,\beta}]\) and are given by \[ \mathcal H^b_{\Omega,\beta} f(x)=\frac{1}{|x|^{n-\beta}}\int_{|t|<|x|}(b(x)-b(t))\Omega(x-t)f(t)\,dt. \] The authors observe that, as in the case of the classical Hardy operator, the conditions for functions \(b\) to define a bounded commutator on Lebesgue spaces define a class bigger than \(BMO\), is the so called central \(BMO\), denoted \(CMO\). Given \(1<q<\infty\), a function \(f\in L^q_{loc}(\mathbb R^n)\) belongs to \(CMO^q(\mathbb R^n)\) is \[ \sup_{r>0}(\frac{1}{|B(0,r)|}\int_{B(0,r)}|f(x)-f_B|^q\,dx)^{1/q}, \] where \(f_B=\frac{1}{|B(0,r)|}\int_{B(0,r)}f(y)\,dy\). Their main result establishes the boundedness of \(\mathcal H^b_{\Omega,\beta}\) from \(L^{p_1}(\mathbb R^n)\) to \(L^{p_2}(\mathbb R^n)\) for functions \(b\in CMO^{q}(\mathbb R^n)\), where \(q\) depends on the parameters on the spaces and the integrability of the kernel. In fact, the proof is done not only for Lebesgue spaces, but for homogeneous Herz spaces \(K^{\alpha, p}_q(\mathbb R^n)\). Also, a similar result is shown for central Morrey spaces, defined by the condition \(\sup_{r>0}(\frac{1}{|B(0,r)|^{1+\lambda q}}\int_{B(0,r)}|f(x)|^q\,dx)^{1/q}\), by replacing the central \(BMO\)-space by a certain modification, denoted by \(CMO^{q,\lambda}(\mathbb R^n)\) and defined in a similar fashion.