Let \(\mathbb R^n\), \(n\geq2\), be the \(n\)-dimensional Euclidean space and \(S^{n-1}\) be the unit sphere in \(\mathbb R^n\) with area element \(d\sigma(x')\) on \(S^{n-1}\). Let \(\Omega(x)|x|^{-n}\) be a homogeneous function of degree \(-n\) on \(\mathbb R^n\), with \(\Omega\in L^1(S^{n-1})\) and \(\int_{S^{n-1}}\Omega(x')\,d\sigma(x')=0\), where \(x'=x/|x|\) for any \(x\neq0\). Let \(P(y)=\bigl(P_1(y),\ldots,P_m(y)\bigr)\) be an \(m\)-tuple of real-valued polynomials on \(\mathbb R^n\), Let \(\Delta_\gamma(\mathbb R_+)\) denote the collection of all measurable functions \(h:[0,\infty)\to \mathbb C\) satisfying \(\sup_{j\in\mathbb Z}\bigl(\int_{2^j}^{2^{j+1}}|h(t)|^\gamma dt/t\bigr)^{1/\gamma} 0\). The authors show the following: Let \(h\in\Delta_\gamma(\mathbb R_+)\) for some \(1<\gamma\leq \infty\). Assume that \(\Omega\in L^q(S^{n-1})\) for some \(1