Maximal functions and Hilbert transforms along variable flat curves
Let \(S(x,y)\) be a suitable real-valued function on \({\mathbb R}^2\) vanishing on the diagonal. For \(f\in C^\infty_0({\mathbb R}^2)\), define \[ {\mathcal M}f(x)=\sup_{0<h<\infty}\frac 1h\left|\int^h_0 f(x_1-t,x_2-S(x_1,x_1-t)) dt\right| \] and \[ {\mathcal H}f(x)=\text{ p. v.}\int^\infty_\infty f(x_1-t,x_2-S(x_1,x_1-t)) \frac {dt}t. \] The authors establish the \(L^p({\mathbb R}^2)\) boundedness for \({\mathcal M}\) with \(1<p\leq\infty\) and for \({\mathcal H}\) with \(1<p<\infty\), via the \(L^p({\mathbb R}^2)\) estimates for certain singular integral operators with oscillatory terms.
scientific article; zbMATH DE number 1489489