The commutator of the Bochner-Riesz operator
Let \(\alpha\in\mathbb{R}\) and \(T^\alpha\) be the Bochner-Riesz operator defined by \[ (T^\alpha f)^\land(\xi)=(1-|\xi|^2)^\alpha_+\widehat f(\xi), \] where \(\widehat f\) is the Fourier transform of \(f\) on \(\mathbb{R}^n\) and \(x_+= \max\{x,0\}\) for \(x\in\mathbb{R}\). For a positive integer \(k\) and \(b_1, \cdots, b_k\in\text{BMO}(\mathbb{R}^n)\), the authors defined the commutator \(T^\alpha_{b_1,\cdots,b_k}\) by \[ T^\alpha_{b_1,\cdots,b_k}f(x)=T^\alpha\Biggl( \prod^k_{j=1} (b_j(x)-b_j(\cdot))f(\cdot)\Biggr) (x) \] and then proved that \(T^\alpha_{b_1,\cdots,b_k}\) is bounded on \(L^p(\mathbb{R}^n)\) with the norm no more than \[ C(n,p,k)\prod^k_{j=1}|b_j|_{\text{BMO}} \] if \(n=2,\;0<\alpha<1/2\) and \(4/(3+2\alpha)<p< 4/(1-2\alpha)\) or \(n=3,\;(n-1)/(2n+2)<\alpha<(n-1)/2\) and \(2n/(n+1+2\alpha)<p<2n/(n-1-2\alpha).\)
scientific article; zbMATH DE number 927279
scientific article