\(L^p\) boundedness of some rough operators with different weights (Q1396411)

If \(\Omega\) is a homogeneous function of degree zero on \(\mathbb{R}^n\) (\(n \geq 2\)), then the maximal operator \(M_{\Omega}\) and the singular integral \(T_{\Omega}\) are defined, respectively, by \[ M_{\Omega} f(x) = \sup_{r>0} \frac{1}{r^n}\int_{|x-y |<r} |\Omega (x-y) ||f(y) |dy, \] \[ T_{\Omega}f(x)= \text{ p.v. }\int_{R^n} \frac{\Omega (x-y)}{ |x-y |^n} f(y)dy. \] \textit{J. Duoandikoetxea} [Trans. Am. Math. Soc. 336, 869-880 (1993; Zbl 0770.42011)] and \textit{D. K. Watson} [Duke Math. J. 60, 389-399 (1990; Zbl 0711.42025)] proved the following: If \(\Omega \in L^q(S^{n-1}), q >1\), then \(M_{\Omega}\) is bounded on weighted spaces \(L^p (w)\) for \(1<p<\infty\) where \(w\) is in the Muckenhoupt weights class. If \(\Omega \in L^q(S^{n-1}), q >1\), and \(\Omega\) has average zero on \(S^{n-1}\), then \(T_{\Omega}\) is bounded on weighted spaces \(L^p (w)\) for \(1<p<\infty\). The authors prove that \(M_{\Omega}\) and \(T_{\Omega}\) are bounded from \(L^p (v)\) to \(L^p (u)\) for some suitable weights pair \((u,v)\) defined by \textit{E. T. Sawyer} [Stud. Math. 75, 1-11 (1982; Zbl 0508.42023)].