Multiparameter singular integrals and maximal operators along flat surfaces
As an extension of Calderón-Zygmund theory, singular Radon transforms and maximal functions along the appropriate submanifolds have been intensively studied. \textit{M. Christ, A. Nagel, E. M. Stein} and \textit{S. Wainger} [Ann. Math. 150, 489--577 (1999; Zbl 0960.44001)] showed their \(L^p\) boundedness under some curvature (or ``finite type'') condition. An interesting problem is to establish the \(L^p\) theory for flat manifolds, which are in lack of the finite type condition. \textit{H. Carlsson, M. Christ, A. Córdoba, J. Duoandikoetxea, J. L. Rubio de Francia, J. Vance, S. Wainger} and \textit{D. Weinberger} [Bull. Am. Math. Soc. 14, 263--267 (1986; Zbl 0588.44007)] considered \(L^p\) estimates for maximal functions and Hilbert transforms along flat curves in \(\mathbb{R}^2\). \textit{A. Seeger, S. Wainger, J. Wright} and \textit{S. Ziesler} [J. Geom. Anal. 15, 477--498 (2005; Zbl 1088.42007)] considered singular integrals associated with higher dimensional flat submanifold of the form \(\{ (t, \gamma ( | t | )) : t \in\mathbb{R}^n \}\). \textit{A. Carbery, S. Wainger} and \textit{J. Wright} [Duke Math. J. 101, 499--513 (2000; Zbl 0959.42006)] have found the necessary and sufficient condition for the \(L^p(\mathbb{R}^3)\) boundedness of the double Hilbert transforms associated with polynomial surfaces. The authors consider the \(L^p\) boundedness of the double Hilbert transforms along flat surfaces without finite type conditions. They consider model surfaces of the form \(\{ (t, \Gamma ( t )) : t \in \mathbb{R}^2 \}\) where \(\Gamma (t_1, t_2) = \gamma_1(t_1) \gamma_2(t_2)\). The double Hilbert transform along this surface is defined by \[ Hf(x) = \iint_{\mathbb{R}^1 \times \mathbb{R}^1} f( x_1 - t_1 , x_2 - t_2, x_3 - \gamma_1(t_1) \gamma_2(t_2)) \frac{dt_1}{t_1}\frac{dt_2}{t_2}. \] They prove the following. Suppose that \(\gamma_i\) is a convex function on \([0, \infty)\) such that \(\gamma_i(0) = \gamma'_i(0) =0\) for each \(i=1,2\). If each \(\gamma_i\) is extended as an even function in \(( - \infty, \infty)\), then \(H\) is bounded on \(L^p(\mathbb{R}^3)\) for \(1<p<\infty\). For the proof they deal with the next maximal function \[ Mf (x) = \sup_{J \in Z^2} | M_J^{1,2}*f(x) |, \] where the Fourier transform of \(M_J^{1,2}\) is given by \[ \widehat{M_J^{1,2}}(\xi)= \iint e^{ i [ \xi_1 2^{j_1} t_1 + \xi_2 2^{j_2} t_2 + \xi_3 \gamma_1(2^{j_1}t_1) \gamma_2(2^{j_2}t_2)]} \varphi(t_1) \varphi(t_2)dt_1 dt_2 \] for \(J = (j_1, j_2)\) and \(\xi = ( \xi_1, \xi_2, \xi_3)\) and \(\varphi\) is a nonnegative function such that \(\varphi \in C^{\infty}_{0}[1/2,1]\) and \(\int \varphi dt =1\). They also study the \(L^p\) boundedness of multi-parameter maximal operator and singular integrals associated with the surface \(\{ (t, \Gamma (t) ) : t=(t_1, \dots t_n) \in \mathbb{R}^n \}\).
scientific article; zbMATH DE number 5509273
scientific article