Weak estimates of singular integrals with variable kernel and fractional differentiation on Morrey-Herz spaces (Q680369)

Summary: Let \(T\) be the singular integral operator with variable kernel defined by \(T f(x) = \text{p.v.} \int_{\mathbb{R}^n}(\Omega(x, x - y) / |x - y|^n) f(y) \mathrm{d} y\) and let \(D^\gamma\) (\(0 \leq \gamma \leq 1\)) be the fractional differentiation operator. Let \(T^\ast\) and \(T^{\sharp}\) be the adjoint of \(T\) and the pseudoadjoint of \(T\), respectively. In this paper, the authors prove that \(T D^\gamma - D^\gamma T\) and \((T^\ast - T^{\sharp}) D^\gamma\) are bounded, respectively, from Morrey-Herz spaces \(M \dot{K}_{p, 1}^{\alpha, \lambda}(\mathbb{R}^n)\) to the weak Morrey-Herz spaces \(W M \dot{K}_{p, 1}^{\alpha, \lambda}(\mathbb{R}^n)\) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product \(T_1 T_2\) and the pseudoproduct \(T_1 \circ T_2\) are also given.