A Littlewood-Paley inequality for the Carleson operator.
Like as the Carleson operator \(Cf(x)=\sup_{\xi\in \mathbb R}| \int_{\mathbb T}(x-t)^{-1}e^{-i\xi t}f(t)\,dt| \), the authors give the following: Assume that \(k\in C^2(\mathbb R\setminus \{0\})\) satisfies \(| k^{(j)}(x)| \leq A_j/| x| ^{j+1}\), \(j=0,1,2\), \(k(-x)=k(x)\) and \(k(x)=0\) for \(| x| >2\). Then for the operator \(Sf(x)\) defined by \[ Sf(x)=\sup_{\xi\in \mathbb R}\left | \int_{\mathbb T}k(x-t)e^{-i\xi t}f(t)\,dt\right| , \] there exists \(C_p>0\) such that \[ \| Sf\| _{L^p(\mathbb T)}\leq C_p\| f\| _{L^p(\mathbb T)}, \quad 1<p<\infty. \] As a corollary, they give a Littlewood-Paley type inequality: Let \(n(x)\) be a real-valued measurable function on \(\mathbb T\), and \[ C_jf(x)=\int_{\mathbb T}(x-t)^{-1}\psi(2^j(x-t))e^{-in(x)t}f(t)\,dt,\quad j=0,1,\dots, \] where \(\psi\in C_0^\infty(\mathbb R)\) is even, \(\roman{supp}\,\psi\subset \{1/2\leq | x| \leq 2\}\) and \(\sum_{-\infty}^{\infty}\psi(2^jx)=1\) for \(x\neq0\). Then there exists \(C_p>0\) such that \[ \left\| \left(\sum_{0}^{\infty}| C_jf| ^2\right)^{1/2}\right\| _{L^p(\mathbb T)} \leq C_p\| f\| _{L^p(\mathbb T)},\quad 1<p<\infty. \] They also give a generalization to the \(n\)-dimensional case, \(A_p\)-weighted estimates and vector valued estimates.
scientific article; zbMATH DE number 1539359