Multi-frequency Calderón-Zygmund analysis and connexion to Bochner-Riesz multipliers (Q2875844)

Let \(\Theta:=(\xi_1,\,\ldots,\,\xi_N)\) be a collection of \(N\) frequencies of \(\mathbb{R}^n\). Then an \(L^2(\mathbb{R}^n)\)-bounded linear operator \(T\) is said to be a Calderón-Zygmund operator relatively to \(\Theta\), if there exist operators \(\{T_j\}_{j=1}^N\) and kernels \(\{K_j\}_{j=1}^N\) such that {\parindent=6mm \begin{itemize}\item[(a)] \(T=\sum_{j=1}^N T_j\); \item[(b)] for every function \(f\in L^2(\mathbb{R}^n)\) compactly supported and all \(x\in[\mathrm{supp}(f)]^\complement\), NEWLINE\[NEWLINET_j(f)(x)=\int_{\mathbb{R}^n}K_j(x,y)f(y)\,dy; NEWLINE\]NEWLINE \item[(c)] for every \(x,\,y\in\mathbb{R}^n\) with \(x\neq y\), NEWLINE\[NEWLINE\sum_{j=1}^N\left|\nabla_{(x,y)}e^{i\xi_j\cdot(x-y)}K_j(x,y)\right|\lesssim|x-y|^{-n-1}. NEWLINE\]NEWLINENEWLINENEWLINE\end{itemize}} Denote by \(A_p(\mathbb{R}^n)\) with \(p\in[1,\infty]\) and \(RH_s(\mathbb{R}^n)\) with \(s\in(1,\infty]\), respectively, the classical class of Muckenhoupt weights and the reverse Hölder class. Assume that \(\Theta\) is a collection of \(N\) frequencies of \(\mathbb{R}^n\) and \(T\) is an associated multi-frequency Calderón-Zygmund operator. Then, {\parindent=8mm \begin{itemize}\item[(i)] for any \(p\in(1,\infty)\), \(T\) is bounded on \(L^p(\mathbb{R}^n)\) and NEWLINE\[NEWLINE\|T\|_{L^p(\mathbb{R}^n)\rightarrow L^p(\mathbb{R}^n)}\lesssim N^{|\frac{1}{p}-\frac{1}{2}|};NEWLINE\]NEWLINE \item[(ii)] for \(p=1\), \(T\) is of weak-type \((1,1)\) and NEWLINE\[NEWLINE\|T\|_{L^1(\mathbb{R}^n)\rightarrow L^{1,\infty}(\mathbb{R}^n)}\lesssim N^{\frac{1}{2}};NEWLINE\]NEWLINE \item[(iii)] for any \(p\in(1,\infty)\) and \(w\in A_{p/s}(\mathbb{R}^n)\cap RH_{t'}(\mathbb{R}^n)\) with \(s\in(1,p)\) and \(t\in(1,\infty)\), where \(t':=t/(t-1)\), \(T\) is bounded on the weighted space \(L^p_w(\mathbb{R}^n)\) and NEWLINE\[NEWLINE\|T\|_{L^p_w(\mathbb{R}^n)\rightarrow L^p_w(\mathbb{R}^n)}\lesssim N^{\gamma}, NEWLINE\]NEWLINE where \(\gamma:=\frac{tp}{s\min\{2,s\}}+|\frac{1}{2}-\frac{1}{s}|\).NEWLINENEWLINE\end{itemize}}NEWLINENEWLINEFor the entire collection see [Zbl 1285.00036].