Linear and bilinear Fourier multipliers on Orlicz modulation spaces (Q6582379)

The aim of this paper is to investigate the properties with respect to linear Fourier multipliers and bilinear Fourier multipliers from Orlicz modulation spaces to Orlicz modulation spaces.\N\NLet \(m(\xi)\) be a bounded measurable function on \(\mathbb{R}^d\) (resp. \(m(\xi,\eta)\) on \(\mathbb{R}^{2d}\)) such that\N\[\NT_m(f)(x)=\int_{\mathbb{R}^d}\hat{f}(\xi)m(\xi)e^{2\pi i<\xi,x>}d\xi\N\]\N\[\N(\mathrm{resp.}\quad B_m(f_1,f_2)(x)=\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}\hat{f_1}(\xi)\hat{f_2}(\eta)m(\xi,\eta)e^{2\pi i<\xi+\eta,x>}d\xi d\eta),\N\]\Nwhere \(\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi i<x,\xi>}dx\) is the Fourier transform of \(f\). The study of linear Fourier multipliers in the setting of Lebesgue spaces is classical, and the investigation of bilinear multipliers was originated by \textit{R. R. Coifman} and \textit{Y. Meyer} [Lect. Notes Math. 779, 104--122 (1980; Zbl 0427.42006)], and was continued by \textit{L. Grafakos} and \textit{R. H. Torres} [Adv. Math. 165, No. 1, 124--164 (2002; Zbl 1032.42020)], \textit{C. E. Kenig} and \textit{E. M. Stein} [Math. Res. Lett. 6, No. 1, 1--15 (1999); erratum ibid. 6, No. 3--4, 467 (1999; Zbl 0952.42005)], \textit{M. Lacey} and \textit{C. Thiele} [Ann. Math. (2) 146, No. 3, 693--724 (1997; Zbl 0914.46034)] and others.\N\NOrlicz spaces generalize \(L^p\) spaces and contain Sobolev spaces as subspaces. Linear Fourier multipliers and bilinear Fourier multipliers on Orlicz spaces are investigated by \textit{O. Blasco} and \textit{A. Osançlıol} [Math. Nachr. 292, No. 12, 2522--2536 (2019; Zbl 1440.42041)], \textit{O. Blasco} and \textit{R. Üster} [Math. Nachr. 296, No. 12, 5400--5425 (2023; Zbl 1536.43002); Mediterr. J. Math. 21, No. 1, Paper No. 41, 21 p. (2024; Zbl 1548.43003)], and \textit{R. Üster} [Result. Math. 76, No. 4, Paper No. 183, 15 p. (2021; Zbl 1481.43001)].\N\NOn the other hand, modulation spaces arising from the time-frequency analysis were introduced by \textit{H. G. Feichtinger} [Modulation spaces on locally compact abelian groups. Techn. Rep., University of Vienna (1983)], and appear in the theory of pseudo-differential operators.\N\NIn this paper, the authors define Orlicz modulation spaces and investigate the properties of the operators \(T_m\) and \(B_m\) on the Orlicz modulation spaces.\N\NLet \(\Phi\) be a Young function, \(\omega\) a continuous positive function with polynomial growth such that \(\omega(x+y)\leq \omega(x)\omega(y) \quad (x,y\in\mathbb{R}^d)\), and \(L^{\Phi}(\mathbb{R}^d)\) the Orlicz space defined by \(\Phi\) with the Luxemburg norm:\N\[\NN_\Phi(f)=\inf\{\lambda>0:\int_{\mathbb{R}^d}\Phi(\frac{|f(x)|}{\lambda})dx\leq1\}.\N\]\N\(L^\Phi_\omega(\mathbb{R}^d)\) is also defined by the set of all measurable functions \(f\) such that \(f\omega\in L^\Phi(\mathbb{R}^d)\). For \(\Phi_j\) \((j=1,2)\) Young functions, the spaces \(L^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})\) are defined as follows:\N\N\(F \in L^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})=L^{\Phi_2}(\mathbb{R}^d,L^{\Phi_1}(\mathbb{R}^d))\) has the following properties:\N\N\(F\) is a complex-valued function on \(\mathbb{R}^{2d}\), and\N\[\N\xi\longrightarrow\ F(\cdot,\xi)\in L^{\Phi_1}(\mathbb{R}^d)\N\]\Nis a Banach space valued function with the Luxemburg norm \(N_{\Phi_1}(F(\cdot,\xi))\) in \(L^{\Phi_1}(\mathbb{R}^d)\) and\N\[\NN_{\Phi_1,\Phi_2}(F)=N_{\Phi_2}(N_{\Phi_1}(F(\cdot,\xi)))<\infty.\N\]\N\quad For a positive continuous function \(\omega\) on \(\mathbb{R}^{2d}\), \(L^{\Phi_1,\Phi_2}_\omega(\mathbb{R}^{2d})\) is defined by the set of all functions \(F\) such that \(F\omega\in L^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})\) with the norm\N\[\NN_{\Phi_1,\Phi_2,\omega}(F)=N_{\Phi_1,\Phi_2}(F\omega).\N\]\NMoreover, for \(\phi\in\mathcal{S}(\mathbb{R}^d)\) and \(f\in\mathcal{S}^\prime(\mathbb{R}^d), \ V_\phi(f)\) is defined by\N\[\NV_\phi(f)(x,s)=\int f(u)\overline{\phi(u-x)}e^{-2\pi i<u,s>}du.\N\]\N\(\mathcal{W}_d\) is also defined by the set of all positive continuous functions \(\omega\) on \(\mathbb{R}^d\) with polynomial growth such that \(\omega(x+y)\leq\omega(x)\omega(y)\ (x,y\in\mathbb{R}^d)\). Then for \(0\neq\psi\in\mathcal{S}(\mathbb{R}^d)\),\ \(\omega\in\mathcal{W}_{2d}\) and \(\Phi,\ \Phi_1,\ \Phi_2\) Young functions, the Orlicz modulation spaces \(M_\omega^\Phi\) and \(M_\omega^{\Phi_1,\Phi_2}\) are defined by the following:\N\[\NM_\omega^\Phi=\{f\in\mathcal{S}^\prime(\mathbb{R}^d):V_\psi(f)\in L_\omega^\Phi(\mathbb{R}^{2d})\}.\N\]\N\[\NM_\omega^{\Phi_1,\Phi_2}=\{f\in\mathcal{S}^\prime(\mathbb{R}^d):V_\psi(f)\in L_\omega^{\Phi_1,\Phi_2}(\mathbb{R}^{2d})\}.\N\]\N\[\N||f||_{M_\omega^\Phi}=N_{\Phi,\omega}(V_\psi f):\text{ the norm of }f\text{ in }M_\omega^\Phi(\mathbb{R}^d).\N\]\N\[\N||f||_{M_\omega^{\Phi_1,\Phi_2}}=N_{\Phi_1,\Phi_2,\omega}(V_\psi f):\text{ the norm of }f\text{ in }M_\omega^{\Phi_1,\Phi_2}(\mathbb{R}^d).\N\]\NLet \(\Phi_i,\Psi_i\) be Young functions, \(\omega_i\in\mathcal{W}_{2d}\), and \(M_{\omega_i}^{\Phi_i,\Psi_i}(\mathbb{R}^d)\) be the corresponding Orlicz modulation spaces for \(i=1,2,3\). \N\NIn this paper, the authors investigate the properties of the linear operators \(T_m\) from \(M_{\omega_1}^{\Phi_1,\Psi_1}(\mathbb{R}^d)\) to \(M_{\omega_2}^{\Phi_2,\Psi_2}(\mathbb{R}^d)\) and the bilinear operators \(B_m\) from \(M_{\omega_1}^{\Phi_1,\Psi_1}(\mathbb{R}^d)\times M_{\omega_2}^{\Phi_2,\Psi_2}(\mathbb{R}^d)\) to \(M_{\omega_3}^{\Phi_3,\Psi_3}(\mathbb{R}^d)\).