İstanbul University Mathematics Department Seminars

Notes on bilinear multipliers on Orlicz spaces
Alen Osançlıol
Beykent Üniversitesi, Turkey
Özet : Let $\Phi_{1}, \Phi_{2}$ and $\Phi_{3}$ be Young functions and let $L^{\Phi_1} (\mathbb{R}), L^{\Phi_2} (\mathbb{R}),$ and $L^{\Phi_3} (\mathbb{R})$ be the corresponding Orlicz spaces. We say that a function $m(\xi, \eta)$ defined on $\mathbb{R}\times \mathbb{R}$ is a bilinear multiplier of type $\left(\Phi_{1}, \Phi_{2}, \Phi_{3}\right)$ if $B_{m}(f, g)(x)=\int_{\mathbb{R}} \int_{\mathbb{R}} \hat{f}(\xi) \hat{g}(\eta) m(\xi, \eta) e^{2 \pi i(\xi+\eta) x} d \xi d \eta$ defines a bounded bilinear operator from $L^{\Phi_1} (\mathbb{R})\times L^{\Phi_2} (\mathbb{R})\to L^{\Phi_3} (\mathbb{R})$ We denote by $\mathcal{B M}_{\left(\Phi_{1}, \Phi_{2}, \Phi_{3}\right)}(\mathbb{R})$ the space of all bilinear multipliers of type $\left(\Phi_{1}, \Phi_{2}, \Phi_{3}\right)$ and investigate some properties of such a class. Under some conditions on the triple $\left(\Phi_{1}, \Phi_{2}, \Phi_{3}\right)$ we give some examples of bilinear multipliers of type $\left(\Phi_{1}, \Phi_{2}, \Phi_{3}\right)$. We will focus on the case $m(\xi, \eta) = M(\xi - \eta)$ and get necessary conditions on $\left(\Phi_{1}, \Phi_{2}, \Phi_{3}\right)$ to get non-trivial multipliers in this class. In particular we recover some of the known results for Lebesgue spaces. This is a joint work with Oscar Blasco (University of Valencia).
  Tarih : 23.10.2019
  Saat : 14:30
  Yer : İ.Ü. Matematik Bölümü D-II
  Dil : Turkish
  Ek Dosya : Özet