Singular integral operators along surfaces of revolution (Q1856875)

In this paper, the author obtains the boundedness of certain singular integral operators along surfaces with highly singular kernels. The singular integral operator studied in this paper is defined, for \(\alpha,\beta> 0\), by \[ T_{\alpha,\beta}f(x,x_n)= \text{p.v. }\int_{\mathbb{R}^n}f(x- y,x_n- \gamma(|y|)) e^{i|y|^{-\beta}}h(|y|) \Omega(y)|y|^{-n-\alpha} dy, \] where \(x,y\in \mathbb{R}^n\), \(x_n\in \mathbb{R}\), \(\Omega\in L^1(S^{n-1})\) is homogeneous of degree zero and satisfies mean zero over the unit sphere \(S^{n-1}\), \(h\) is a bounded, differentiable radial function, \(\gamma\) is a measurable even function such that \(|\gamma'(r)|\) is increasing on \((\text{supp} \gamma')\cap [0,\infty)\). Assume that either \(h\) is monotone or \(h'\in L^1(\mathbb{R})\) and that \(\gamma'\in L^1(\mathbb{R})\) or \(\gamma\in L^\infty(\mathbb{R})\) and \(\gamma(r)\) is monotone on \([0,\infty)\). The author proves that if \(\Omega\) is a function in the Hardy space \(H^1(S^{n-1})\) then \(T_{\alpha,\beta}\) is bounded on \(L^2(\mathbb{R}^{n+1})\) provided \(0< 2\alpha< \beta\). He also proves that \(T_{\alpha,\beta}\) is bounded on \(L^p(\mathbb{R}^{n+1})\) provided \(\beta/(\beta-\alpha)< p< \beta/\alpha\) with \(0< 2\alpha< \beta\) and that the maximal function \[ M^\gamma g(s)= \sup_{r> 0} r^{-1} \int_{|t|< r}|g(s- \gamma(t))|dt \] is bounded on \(L^p(\mathbb{R}^n)\) for \(1< p< \infty\). Several applications are also given.