Multipliers of multiple Fourier series (Q2714719)

Let \(1< p\leq q<\infty\). A multiple sequence \(\lambda= \{\lambda_k: k\in\mathbb{Z}^n\}\) of complex numbers is said to be a multiplier of Fourier series from \(L_p(\mathbb{T}^n)\) to \(L_q(\mathbb{T}^n)\) if for every function \(f\in L_p(\mathbb{T}^n)\) with Fourier series \(\sum_{k\in\mathbb{Z}^n}\widehat f(k) e^{ikx}\) there exists a function \(f_\lambda\in L_q(\mathbb{T}^n)\) whose Fourier series coincides with the series \(\sum_{k\in\mathbb{Z}^n} \lambda_k\widehat f(k) e^{ikx}\). The set \(m^q_p\) of all such multipliers endowed with the norm NEWLINE\[NEWLINE\|\lambda\|:= \sup\{\|f_\lambda\|_q/\|f\|_p: f\neq 0\}NEWLINE\]NEWLINE is a linear space. The present authors prove lower and upper estimates of this norm, and thereby give necessary or sufficient conditions for a sequence \(\lambda\) to belong to \(m^q_p\).NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].