The boundedness for commutator of fractional integral operator with rough variable kernel (Q1935420)

For \(b\in \text{BMO}(\mathbb{R}^n)\) and \(0\leq \alpha <n\) the commutator of the fractional integral operator \(T_{\Omega,\alpha}\) with rough variable kernel is definedby \[ [b,T_{\Omega,\alpha}]f(x)=\int_{\mathbb{R}^n} \frac{\Omega(x,x-y)}{|x-y|^{n-\alpha}}(b(x)- b(y))f(y)dy. \] The following results are proved in the paper. Theorem 1. Let \(0<\alpha\leq 1/2\) and \(b\in \text{BMO}\). If \(\Omega\in L_{x}^{\infty}(\mathbb{R}^n) \times L_{z^{\prime}}^{q}(\mathbb S^{n-1})\) with \(q> \frac{2(n-1)}{n-2\alpha}\), then there is a constant \(C>0\) such that \[ \|[b,T_{\Omega,\alpha}]f\|_{L^2}\leq C \|\Omega\|_{L_{x}^{\infty}(\mathbb{R}^n) \times L_{z^{\prime}}^{q}(\mathbb S^{n-1})} \|b\|_{\text{BMO}} \|f\|_{\frac{2n}{n+2\alpha}}. \] Corollary 1. Let \(0<\alpha\leq 1/2\). If \(\Omega\in L_{x}^{\infty}(\mathbb{R}^n) \times L_{z^{\prime}}^{q}(\mathbb S^{n-1})\) with \(q>\frac{2(n-1)}{n-2\alpha}\), then there is a constant \(C>0\) such that \[ \|T_{\Omega,\alpha}\|_{L^2}\leq C \|\Omega\|_{L_{x}^{\infty}(\mathbb{R}^n) \times L_{z^{\prime}}^{q}(\mathbb S^{n-1})} \|f\|_{\frac{2n}{n+2\alpha}}. \] It is also proved that in Theorem 1 and Corollary 1 the exponent \(\frac{2(n-1)}{n-2\alpha}\) is sharp.