Rough singular integrals associated to surfaces of revolution (Q2723489)

Let \(n\geq 2\) and \(y\in\mathbb{R}^n\). For the Calderón-Zygmund type kernel \(K(y)= b(|y|)\Omega(y)|y|^{-n}\) the singular integral \(Tf\) studied in this paper is defined by NEWLINE\[NEWLINETf(x,s)= \text{p.v. }\int_{\mathbb{R}^n} K(y) f(x- y,s- \Phi(|y|)) dy,NEWLINE\]NEWLINE where \(b\in L^\infty\); \(\Omega\) is homogeneous of order zero on \(\mathbb{R}^n\), integrable on \(S^{n-1}\) and satisfies \(\int_{S^{n-1}} \Omega(y) d\sigma(y)= 0\); \(\Phi\) is a suitable continuously differentiable function on \((0,\infty)\) such that \(Tf(x,s)\) exists for all \((x,s)\in \mathbb{R}^{n+1}\).NEWLINENEWLINENEWLINEThe main purpose of this paper is to consider the \(L^p\) boundedness of \(T\) when \(\Omega\) is in the Hardy space \(H^1(S^{n-1})\). Define a two-dimensional maximal function \(M_\Phi\), associated to \(\Phi\), by NEWLINE\[NEWLINEM_\Phi f(x_1, x_2)= \sup_k 2^{-k} \int^{2^{k+1}}_{2^k}|f(x_1- t, x_2- \Phi(t))|dt.NEWLINE\]NEWLINE The authors prove that if \(\Omega\in H^1(S^{n-1})\) then \(T\) is bounded on \(L^p(\mathbb{R}^{n+1})\) for \(1< p< \infty\), provided that \(M_\Phi\) is bounded on \(L^p(\mathbb{R}^2)\).