\(L^p\)-boundedness of wavelet multipliers
If \(\varphi\in L^2({\mathbb R}^n)\cap L^\infty({\mathbb R}^n)\) is such that \(\| \varphi \|_2=1\), and \(\sigma\in L^\infty({\mathbb R}^n)\), in this paper the corresponding wavelet multiplier is the linear operator \(P_{\sigma,\varphi}\) such that \[ (P_{\sigma,\varphi}u,v)_2 =(2\pi)^{-n} \int_{{\mathbb R}^n} \sigma(\xi)(u,\pi(\xi)\varphi)_2 (\pi(\xi)\varphi,v)_2\, d\xi\quad (u,v\in\mathcal S), \] where \((\pi(\xi)\varphi)(x)= e^{ix\cdot\xi}\varphi(x)\). The authors give conditions on the symbol \(\sigma\) to ensure that \(P_{\sigma,\varphi}\) is a bounded operator on \(L^2({\mathbb R}^n)\), and on \(L^p({\mathbb R}^n)\) for \(4/3<p<4\).
scientific article; zbMATH DE number 2140986