On modified singular integrals (Q1321761)

A singular integral operator \(Tf(x)=\text{p.v.}\int\Omega (x- y)f(y)dy\), where \(\Omega\) is a homogeneous function of degree \((-n)\) on \(\mathbb{R}^ n\) with zero mean over the unit sphere, is bounded on the weighted Lebesgue space \(L_ p(| x|^ \alpha)\) if and only if \(-n< \alpha< n(p- 1)\) [\textit{E. M. Stein}, Proc. Am. Math. Soc. 8, 250-254 (1957; Zbl 0077.273)]. The author introduces modified singular integral operators which are bounded on \(L_ p(| x|^ \alpha)\) for every \(\alpha< n(p- 1)\), \(\alpha\neq -n-kp\), where \(k\) is a given non-negative integer.