A method for proving \(L^p\)-boundedness of singular Radon transforms in codimension 1
The author considers the following operator: \[ Tf(x,t)=\int_{\mathbb R^n} f(t+S(t,x,y),y)K(t,x,y) dy \] , where \(x,y\in \mathbb R^n\), \(t\in \mathbb R^n\), \(K(t,x,y)\) is a distribution supported on a compact set \(U\) and satisfies \[ |\partial_x^\alpha K(t,x,y)|\leq M|x-y|^{-n-|\alpha|}, \] \[ |\partial_y^\alpha K(t,x,y)|\leq M|x-y|^{-n-|\alpha|}, \] \[ |\partial_t^\alpha K(t,x,y)|\leq M|x-y|^{-n} \] for \(0\leq |\alpha|\leq 1\), \[ \left|\int_{|y-x|=a}K(t,x,y) dx\right|, \left|\int_{|y-x|=a}K(t,x,y) dy\right|\leq Ma \] for \(a>0\), and \(S(t,x,y)\) is a \(C^\infty\)-function satisfying \(S(t,x,x)=0\), \(|\partial_tS|<1/2\), \[ \sup\{|\partial_x^\alpha\partial_y^\beta\partial_t^\gamma S(t,x,y)|;\;(t,x,y)\in U, |\alpha|+|\beta|+|\gamma|\leq d+1\}\leq M \] for some positive integer \(d\), and some curvature condition equivalent to a finite-type condition given by \textit{M. Christ, A. Nagel, E. M. Stein}, and \textit{S. Wainger} [Ann. Math. (2) 150, No.~2, 489-577 (1999; Zbl 0960.44001)]. For the above operator, he gives another proof of the \(L^p\)-boundedness by geometric and analytic methods, without using the method of lifting to nilpotent Lie groups. He discusses a few directions in which generalizations of his arguments might be attempted.
scientific article; zbMATH DE number 1820824
scientific article